Sokolow- Differentiated Instruction

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Citation Sokolow, Karen. 2018. Sokolow- Differentiated Instruction. Master's thesis, Harvard Extension School.

Citable link https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37365369

Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Design and Implementation of a High School Mathematics Class Using Differentiated Instruction

Techniques

Karen Sokolow

A Thesis in the Field of Mathematics and Teaching

for the Degree of Master of Liberal Arts in Extension Studies

Harvard University

This thesis describes the design and implementation of a high school mathematics class using a variety of Differentiated Instruction (DI) techniques. The course offers an alternative to the traditional mathematics curriculum sequencing – starting with Algebra

1, progressing through Geometry, Algebra 2, PreCalculus before terminating with either

Calculus or Statistics – with the goal of maintaining DI as a core concept. Originally, the class was offered as an alternative to a conventional calculation-based approach common in a standard high school mathematics class. Instead, the class presented a project-based interdisciplinary approach to a variety of topics ranging from Number History to Music to Personal Finances. The mathematics department kept three distinct goals in mind when proposing the class. First, we hoped to retain enrollment in mathematics beyond baseline graduation expectations. Second, we hoped to engage the interest of students who do not display overt enthusiasm or aptitude in the established sequence of classes. Third, we hoped to offer a challenge to students who display high mathematical skill or high interest by giving a survey of more advanced topics and the historical, theoretical, practical and creative contexts of mathematics.

In designing this class, I noticed a relative lack of significant DI lesson plans. I wondered if this was a function of the subject. Perhaps mathematics did not lend itself to creative differentiation and meaningful integration of learning modalities. I sought to design a class that would concentrate on differentiation as a core concept and explore the benefits this could give the students. I aimed to provide a framework for meaningful content acquisition in a context that could be applied to both non-traditional and more classically structured classrooms and classes.

In order to integrate the student experience into the in-depth examination of

Differentiated Instruction, I elicited student feedback. I requested that students help determine the structure of the class and examine the success of each class design. I asked them to analyze their thought processes, approaches to the DI methods, their responses to the activities and their interactions with the content. Although I set out to alter the class experience through DI, the act of questioning the students and observing the class produced informative and positive results. By asking the students to examine their response to DI, and encouraging their metacognitive efforts, the students were able to demonstrate a more thorough understanding of content than I had anticipated. The focus on DI altered my process as a teacher but the process of asking for student feedback and preferences altered the student experience. Frontispiece

Fibonacci Spiral with horse chestnut. Original artwork and inspiration for class project.

Dedication

To my students who inspire me to be a better teacher.

To Tim who inspires me to be a better person.

Acknowledgments

Thank you to my school community for supporting this project and their faith in my creative vision for the potential of mathematics.

Thank you to Andy Engelward for gently and consistently encouraging me to keep going.

Thank you to my closest friends and family for providing encouragement and support, my partner for unwavering love and a multitude of dinners and the women in my life who, nevertheless, persist.

Table of Contents

Abstract ...... iii

Frontispiece ...... v

Dedication ...... vi

Acknowledgments...... vii

List of Tables ...... xi

List of Figures ...... xii

Chapter I Introduction ...... 1

Inspiration and Thesis Project ...... 2

Chapter II Theoretical Background ...... 7

Multiple Intelligences ...... 8

Learning Styles ...... 12

Differentiated Instruction ...... 14

Exploring Meaningful Differentiation in Mathematics ...... 21

Chapter III Course Design and Execution ...... 25

Overview ...... 25

Timeline of Class ...... 28

Chapter IV Sample Lessons with Discussion ...... 33

Lesson 1: Information Acquisition ...... 33

Lesson 2: Summative Assessment ...... 37

Transition Between Units ...... 42 Lesson 3: Formative Assessment ...... 45

Lesson 4: Information Acquisition ...... 51

Conclusion ...... 57

Chapter V Reflection and Analysis...... 58

Focus on Differentiated Instruction in the Classroom ...... 58

Obstacles and Challenges ...... 65

Role of Technology...... 67

Applications in Other Classrooms ...... 72

Future Considerations for the Class ...... 77

Conclusion ...... 78

Appendix 1 Lesson 1: Information Acquisition...... 80

Changing Bases: Template Notes and Individual Work ...... 80

Appendix 2 Lesson 2: Summative Assessment Text Resources ...... 85

Preparation for Summative Assessment- Study Guide ...... 85

Summative Assessment- Traditional Style Test ...... 86

Summative Assessment- Project Text ...... 88

Appendix 3 Lesson 3: Formative Assessment Student Directions and Assignment ...... 89

Project Text- Introduction to Assignment...... 89

Appendix 4 Lesson 4: Information Acquisition Notes Template ...... 91

Appendix 5 Text of Student Surveys ...... 93

Differentiated Instruction as it pertains to Content Acquisition and Instruction

Technique ...... 93

Differentiated Instruction as it pertains to Assessment ...... 94 Differentiated Instruction as it pertains to Confidence Level and Mood ...... 95

Differentiated Instruction as it pertains to Content of Course ...... 96

Appendix 6 Student Responses to Selected Survey Questions ...... 99

Student Response to Preferred Assessment Style ...... 99

Student Response to Question: What makes a “good” project? ...... 101

An “A” Project: ...... 102

A “B” Project: ...... 103

A “C” Project: ...... 103

A “D or F” Project: ...... 104

Other Responses...... 104

Responses to the questions about confidence and concern with assessments .....105

Student Response to Preferred Classroom Instruction...... 107

References ...... 108

List of Tables

Table 1. Topics for Consideration ...... 29

Table 2. Student Generated Ideas for Topics of Study 2016-2017 ...... 30

Table 3. Breakdown of Student Selection for weighted test/project ...... 41

Table 4. Group Work Assignments and Difficulty Levels...... 50

Table 5. Notes Template ...... 92

List of Figures

Figure 1. Scope and Sequence of Mathematics Classes ...... 27

Figure 2. Differentiated Instruction for Content Acquisition...... 36

Figure 3: Student Selection of Course of Study – First Choice ...... 43

Figure 4: Student Selection of Course of Study – Second Choice ...... 43

Figure 5: Student Selection of Course of Study – Third Choice ...... 44

Figure 6. Student Response to Assessment Style ...... 99

Figure 7. Student Response to Preferred Instruction Style ...... 107

Chapter I

Introduction

Each math teacher, at some point, has been confronted by a version of the question, “When are we going to use this in real life?” Teachers may give answers that range from practical (money and cooking with recipes) to professional (engineers and accountants), topical (this skill informs the next skill) to existential (this skill may not be useful in day-to-day life, but the thought process is important). Each teacher most likely tailors the answer to the group of students, the level of the class, the type of relationship between the students and teacher and even the time of year.

Each of these answers, though, fails to address an underlying issue. Students who possess an intrinsic love of mathematics or a propensity for the topic do not usually question the “Why” of mathematics. Students ask this question in an attempt connect with the material. They may lack either an aptitude or an interest in mathematics. For them, mathematics is a sterile, unengaging or irrelevant field of study. It is possible that these students perceive mathematics as a discipline for other people because they do not see their interests or identities reflected in the curriculum. Many students understand that mathematics is important for a well-rounded education but form only a passing understanding of the material. Some students engage in the material only enough to pass the course, have it look good on a transcript and get into a good school. In anecdotes, students will often choose an Honors or Accelerated class, not because they enjoy, or even truly understand the intricacies of the material, but because of external pressures of the realities of the pursuit of higher education.

Teachers must, then, consistently ask the question, how can I elicit engagement, curiosity and interest from the most disenfranchised student? How can I, as a leader and guide in the classroom, draw forth the best performance from the most reluctant of students? Attempting to heighten engagement, interest and long-term retention of the material, educators reach to a variety of methods to tailor the learning experience to the individual students. Project-based learning, flipped classrooms, growth mindset research,

Multiple Intelligences and learning modalities, and Differentiated Instruction are all topics that cycle in and out of holding a central role in education best practices pedagogy.

Technological integration continues to gain importance as both a skill to teach students and a tool to use in the classroom. With each of these pedagogies firmly established and well-researched, it is sometimes difficult for the average teacher to choose the right method for the students, the classroom and the school.

Inspiration and Thesis Project

The core philosophies and support structures within the school most readily and directly support classroom integration of Multiple Intelligences (MI), learning modalities and Differentiated Instruction (DI). Howard Gardner’s theory of Multiple Intelligences posits that intelligence has a multitude of manifestations that most classrooms, and certainly most assessments, access. For teachers, MI can be a source of inspiration for creative and innovative lesson planning. It is, though, difficult to push the boundaries of creativity while constrained within the traditional sequencing of mathematics curricula.

Explicitly address learning styles, student preferences for the method of information acquisition, may increase overall understanding by a student, but is challenging to integrate in meaningful ways into a subject with the abstract nature of mathematics

Differentiated Instruction, as a framework of best practices and pedagogical ideas for scaffolding both instructional and assessment practices, can offer the individual student a variety of instructional strategies in order to find the best method for the individual.

Ideally, it is the goal of DI that students access information at their readiness level, content fluency level and then best demonstrate their content acquisition in a way that is appropriate for their cognitive framework. It is, however, an incredibly broad category.

There is no centralized theory of DI and little consensus about how to best implement practices across disciplines, age groups and curricula. Resources that highlight DI in a math classroom specifically can be problematic to interpret. Some resources share strategies that are completely separate from the content of the classroom while others have a multitude of resources for younger children or first-time mathematics learners, but few for higher level math courses or older students. Still other lessons that differentiate according to learning styles suggest elements are non-essential and non-integral to the lesson. Although I worked in a place the embraced both MI and DI theories, I struggled to find a meaningful way to use MI as a creative resource and DI as a fully cohesive classroom ethos.

Initially, I conceptualized this thesis as an exploration of MI and DI theories to increases student access, understanding and engagement in mathematics. In principle, MI expands the ideas of intelligence and cognition, learning styles allow teachers to access their students cognitive process while DI is the catch-all term for a variety of guidelines governing information acquisition and assessment of knowledge. I had high hopes that

MI, learning styles and DI would offer some creative and fresh approaches to students who struggle with traditional mathematics and calculation courses or those students who

are not intrinsically motivated by the discipline of mathematics. Resources on MI, though, consistently disappointed me. Lesson plans were rarely for older or more mature students. I attended conferences, reading books and articles and through many, many, conversations with other math teachers at the school and searched for MI and DI resources that added inherent value to the class, were age appropriate and were novel ideas. While several sources had some options (Tomlinson & Imbeau, 2010, p. 245-250), math was at times neglected, at times combined with sciences (Tomlinson & Stickland,

2005) and paled in detail and creativity to sample lessons and ideas in the humanities. As

I explored differentiation with technology, I found a multitude of ways to differentiate homework and problem sets. At the National Council of Teachers of Mathematics annual conference in 2015, I learned about math games, math apps, digital platforms for assigning homework, video tutorials. It was overwhelming. Some applications of technology impressed me quite a bit. I left, though, with the same dissatisfaction. Either the technology added flair, but not substance, to the classroom, or required a significant amount of knowledge of computer language or software specific training. There seemed to be few options in the middle ground. I started to wonder, was mathematics a categorically different discipline? Did it defy foundational and creative differentiation because of its abstraction from the student experience?

As I searched through the theories of MI, learning styles and DI, looking for meaningful pathways of differentiating instruction, it occurred to me that I might be looking in the wrong place. The content, at least in the established high school mathematics curriculum, is fairly static. The students, however, are constantly changing.

The most differentiated aspect of the classroom was not necessarily the content, but the

student body itself. Although students may resemble each other, like snowflakes, no two students are alike. Each comes from a unique background, has a unique mindset and cognitive framework, and possess a unique set of intelligences, strengths and weaknesses. Perhaps I was searching to differentiate the content instead of differentiating for the students. The mind shift was subtle, but provoking.

I conceptualized a class that approached mathematics, not from a calculation- based perspective, but from a personal perspective. Instead of designing a class with the content being the central tenet, the what of mathematics, I began with a different set of premises, the who, how and why, of mathematics. I hoped that approaching mathematical content from a biographical, historical, philosophical, artistic, practical, etc. perspective, would connect the students with the mathematics on a personal and emotional level.

Several lectures at the NCTM Annual Conference in Boston, 2015, explored this work.

Kaci Elder, a poet and story-teller, had paired with historians and mathematicians to explore social justice and the role of personal and cultural narrative in mathematics

(Elder, 2015). Perhaps, allowing the lived experience of the student to take a central role in the class would allow for a more creative differentiation that would be positively value-additive.

This thesis examines methods of Differentiated Instruction and explores ways to design a class that meaningfully intertwines DI and mathematics as a discipline. These topics will be explored through an elective class I designed. The class decentralizes calculation-based content and, instead, contextualizes the human experience of mathematics. Once a relationship between the students and the subject is established using differentiation techniques, then students will be introduced to more traditional

mathematical concepts. I drew heavily on MI theory and interdisciplinary education, using biography, philosophy, art, music, history etc. to foster student interaction. After the student understands the greater context, the abstract mathematical thought can be given as a set of tools, in greater or lesser amounts as the student is cognitively able, to explore the specific mathematical content. If the calculation and heavy abstraction of concepts was the alienating factor, shifting the focus to context would mimic student comfort zone and through explicit differentiation, encourage them to reach beyond. The research of pedagogies, class design with lesson plans, student observations and retrospective analysis of the class design follows.

Chapter II

Theoretical Background

Although the normative cultural narrative holds a singular view of intelligence, that it is quantifiable and largely static, several other theories call into question this simple story. The Global Capacity View, introduced by David Weschler in the 1940s, promotes the idea that intelligence is the aggregation of abilities that enable an individual to act purposefully, think rationally and interact effectively with the environment

(Weschler, 1940). Rather than treat intelligence as an isolated measurable category,

Weschler theory treats intelligence as the combination of several factors including personality and the environment. Unlike the Stanford Binet IQ test which gives a lone, cumulative score, Weschler’s test measures four categories and then includes an aggregate score and a description of how the categories interact. The four categories measured in the Weschler Intelligence Scale are: Verbal Comprehension, Perceptual

Reasoning, Working Memory and Processing Speed. The test is designed for both children and adults and commonly used today.

After the Global Capacity View came several other theories that called into question a generalized singular view of intelligence. The Triarchic Theory, for example, offered yet another theory of intelligence. Robert Sternberg, in the 1980s, characterized intelligence with three distinct components, rather than a single ability or disability.

Sternberg distinguished between analytical skills, creativity and practical skills.

According to Sternberg, traditional intelligence tests often ignored creative or practical skills. He was also one of the first theorists to draw a distinction between the performance components of intelligence, the knowledge acquisition components and the

metacomponents, which he described as the processes that are needed to plan or make decisions (Sternberg, 1988). Unlike Weschler, who developed a measurement system and widely used test, Sternberg’s view of intelligence is not easily quantifiable but has codified indicators to evaluate.

Weschler’s Global Capacity View and Sternberg’s Triarchic Theory, as well as numerous other psychologists and scientists, set the stage for Howard Gardner’s Theory of Multiple Intelligences by challenging the ruling idea that intelligence was static, singular and easily quantifiable. They set the stage for a new way of thinking about knowledge, intelligence and content acquisition.

Multiple Intelligences

Howard Gardner, in the late 1980s and early 1990s, posited that the traditional

Stanford-Binet Intelligence Quotient (IQ) Test only addressed certain aspects of intelligence. It focused heavily on linguistic proficiency and logical/spatial competency.

The traditional IQ test completely ignored a person’s musical prowess or ability to learn physical activities quickly and with a great deal of competency (Gardner 1973). Though commonly labeled as talents or aptitudes, Gardner recategorized certain aspects of the mind as intelligences. Gardner posited that there were seven, later amended to eight, distinct ways that a person could be intelligent. The primary basis of intelligence, for

Gardner, centered around a person’s ability to solve problems and interact with the world.

This ability had to arise out of a predictable and, preferably, historical or evolutionary context. Inclusion criteria was quite strict. Each intelligence had to fulfill the following eight criteria:

1. The intelligence must be potentially isolatable vis brain damage or mental

illness.

2. The intelligence must demonstrate the existence of extremes. For example,

tone-deafness or musical savants would demonstrate the extreme examples of

musical intelligence.

3. The intelligence must have a distinct development with describable end-state

expertise or performance

4. The intelligence must have an evolutionary explanation or history

5. The intelligence must be supported measurable findings

6. The intelligence must derive supports from experiments

7. The intelligence is unique and identifiable

8. The intelligence must be susceptible to encoding in a symbol system, usually

for the purpose of transmitting knowledge. While this is most commonly

interpreted as being linguistic in nature, musical notes or mathematical

symbols would also fulfil this criterion as they are symbolic encodings within

the discipline. (Gardner, 1983)

Gardner’s original seven intelligences were Linguistic, Logical-Mathematical,

Spatial, Body-Kinesthetic, Musical, Interpersonal and Intrapersonal. Natural Intelligence was proposed codified and added approximately fifteen years later (Gardner 1999).

Gardner later opened up the possibilities of additional intelligences, including the commonly added, but highly controversial Existential Intelligence. Gardner hesitated to officially add it as an intelligence because he questioned its place in classrooms and worried about how the measurement of a religious and philosophical presence would be

received. Over time Gardner has explored the idea of other intelligences including cooking, humor, pedagogical and sexual, but none lack the strength of the original eight

(Gardner, 2016). Each intelligence is marked by a precise set of intrinsic rules for assessing the world and learning from it. Certain intelligences tend to impact the way a person internalizes information while others tend to impact the ways in which a person presents or conveys information. Each person possesses all intelligences, although not evenly, and a person’s intelligence map can adapt and change over time (Project Zero,

2016, pg 27-30).

Linguistic Intelligence is marked by the capacity to use language effectively and efficiently in either an oral or written fashion. People who demonstrate linguistic intelligence often pursue careers as poets, playwrights, journalists, storytellers, orators, politicians or motivational speakers. It should be noted that linguistic intelligence may, although is not required to, be accompanied by quick foreign language acquisition and fluency.

Logical-mathematical Intelligence is marked by the capacity to use numbers effectively and efficiently. It is also indicated by an attention to logical patterns and relationships or abstractions. People who demonstrate this intelligence excel at categorization, classification, calculation, inference and generalization. Accountants,

Scientists, Computer Programmers, Statisticians etc. are all careers that tend to attract people who demonstrate logical-mathematical intelligence. It should be noted that most standardized tests, including the Stanford-Binet IQ Test, heavily measure logical- mathematical intelligence.

Spatial Intelligence is marked by the ability to perceive and manipulate the visual- spatial world accurately. People who demonstrate this intelligence may be able to graphically represent spatial ideas or orient oneself in a spatial manner. Professions that require spatial intelligence range from hunter to interior decorator, wilderness guide to artist, scout to architect.

Body-Kinesthetic Intelligence is marked by the ability to use one’s whole body to express ideas or the ability to use one’s body to transform the world around them. This intelligence requires dexterity, speed, strength, strong proprioceptive tendencies and tactile fluency. People who exhibit Body-Kinesthetic Intelligence may have professions that range from actor to surgeon.

Musical Intelligence is marked by the capacity to perceive, transform and alter/manipulate musical forms. This intelligence is marked by a sensitivity to tone, timbre, pitch, rhythm, melody, and more. People who possess Musical Intelligence may be performers, musicians, dancers or critics. Importantly, people who like music may not possess strong Musical Intelligence. An interest in a topic is not the same as an intelligence in the topic.

Interpersonal Intelligence is marked by the ability to perceive and then change behavior based on others’ mood, affect, body language, tone and other non-verbal communication. People who exhibit interpersonal intelligence are sensitive to small body cues and respond in pragmatic ways.

Intrapersonal Intelligence is marked by a strong sense of self and the ability to adapt to new information or situations based on self-assessment. People who exhibit a strong Intrapersonal Intelligence have a high capacity for self-awareness and self-esteem.

Natural Intelligence is marked by tendencies to recognize and classify the environment around the person. This includes a sensitivity to natural phenomena, like local flora and fauna, or in cases of urban setting, man-made phenomena like architecture or automobiles (Gardner 1983).

Existential or Spiritual is marked by sensitivity and ability to process questions of existence and purpose. People who exhibit this intelligence engage in reflective and deep thinking easily and are comfortable with abstract thought. Scientists, philosophers and theologians often possess Existential Intelligence. It should be noted that this

Intelligence is only included as an “official intelligence” sometimes and Howard Gardner refers to it as the 81 intelligence. It fulfills most of the criteria to be an accepted 2 intelligence, although it often lacks support from measurable findings, as most of the measurement is being offered by the people who possess this intelligence. Gardner has, so far, avoided to definitively commit to the inclusion of Existential, or any other,

Intelligence (Project Zero, 2016, pg 26-27).

The Theory of Multiple Intelligences provides a framework for understanding the multifaceted nature of intelligence. Rather than issuing a single number that encompasses a directly measurable problem-solving set, MI gives individuals a collection of strengths and aptitudes.

Learning Styles

It is important to note that Multiple Intelligence provides only the cognitive structure with which a person experiences the world. It does not directly provide the

tools with which to teach a student. Often, the theory of multiple intelligences is confused with learning styles.

A learning style is a consistent or persistent preference for perceiving, thinking about and organizing information. Learning styles are preferences, not intelligences or behaviors (Snowman, 2009 pg. 120). A student may have learning styles that are sensory in nature or cognitive in nature. Some students may prefer quiet environments, auditory input and highly organized structures. Other may have no organizational preferences, but prefer visual input in group settings (Snowman, 2009 pg. 120-122). Commonly, learning styles are divided into visual, auditory, read/write and kinesthetic (VARK). Classroom activities may be designed to correlate to a learning style but not an intelligence. Much of the confusion between the two lies in the high overlap of terminology. A student who exhibits high Body-Kinesthetic Intelligence may not necessarily be able to optimally process mathematical information with a kinesthetic learning style. Best practices tend to be to teach the same information in a variety of styles in order to allow each student equal access to the attention. Anne Larsen, speaker and the Landmark School summer outreach program, stressed the importance of learning style, “If you have a student who has a high preference for a modality and do not teach in that mode, their strength becomes a distraction.” Incorporating multisensory learning activities into a classroom may allow students the opportunity to use a multiplicity of intelligences but the two are distinct pedagogical tools.

There is some debate about the validity of learning styles. Teachers have no doubt that a multifaceted approach to classroom content fosters broader understanding, but scientists and scholars are not as sure of this approach. First, designing experiments

to test learning styles would be quite difficult as, often, there is overlap in a single lesson or within a single student. Second, learning styles are often culturally dictated and formed in the early days of schooling. Finally, grouping students according to style can actually be more detrimental to a holistic experience than helpful (Finley, 2015 pg. 1-4).

On the flip side, though, educators and educational theorists argue that learning styles change over time and engagement is the key to understanding. Additionally, the lack of quality large-scale studies makes any data driven discussion of learning styles extremely difficult (Finley, 2015 pg. 5-9).

Overall, although the science surrounding learning styles is murky, all educators and scholar alike agree that engagement is the key to success. Using multisensory activities and learning styles, like VARK, in the classroom can only enhance the totality of educational experience for the student.

Differentiated Instruction

Differentiated Instruction came into popularity between 1970 and 2000, concurrent with the reassessment of traditional ways of measuring intelligence

(Snowman, 2009 pg. 112-113). Differentiated Instruction (DI) is a loosely organized set of guidelines that describes a practice of using various learning materials, instructional methods, assignments and assessments to accommodate differences among students

(Snowman, 2009 pg. 108). Because students vary in their ability and disability, learning styles and preferences, prior knowledge and readiness, not every student comes into a classroom the with the same educational framework. Because students differ in their gender and sexuality, ethnicity and race, cultural background, linguistic fluency or competency, and socio-economic status, not every student comes to the classroom with

the same socioemotional framework (Snowman 2009, pg. 211). Optimal educational experiences start with instruction meeting the student in their comfort zone and then allow the student to grow into the knowledge in the way that is the most effective for their cognitive framework. Since each student is unique, it would stand to reason that each student needs a unique approach. DI offers a set of tools to give the unique approach.

It is important to note that DI is not a single cohesive theory or notion. Rather it is a collection of pedagogies and best practices. For this reason, it has a somewhat amorphous set of ideals that shift over time and are updated with current educational research. DI practices strive to provide teachers with the research and tools to create the best educational opportunities for the student in a student-centered and responsive classroom. This responsiveness is both a benefit and detriment of DI. The lack of central cohesion gives the theory agility. As educational needs change, DI practices can also change, and quickly. Teachers can select the DI frameworks that work best in their school environment and with their school culture. Instruction and assessment can be differentiated immediately and as needed, without the slow consideration to a rigid set of founding ideals. A DI based environment can incorporate new research or social understandings promptly. This agility, however, presumes that the educator is responsive to and aware of national and cultural shifts in education policy. The rapid implementation also presumes a school environment that favors and fosters and can afford creative implementation of educational pedagogies.

A lack of clear and simple core tenets also leads to confusion about what is and what is not differentiation. Does differentiation occur at a human level? At an

organizational level? At a curriculum level? At a grading level? The answer is all of the above, which yields a strange paradox wherein if everything is Differentiated Instruction, nothing is (Bird 2005). Rick Wormeli, in Fair Isn’t Always Equal: Assessing and

Grading in a Differentiated Classroom, begins with snapshots of a differentiated classroom.

“Some students have preferential seating because of attention problems.

The teacher moves physically closer to some students, using proximity to him or her to keep them focused.

Desks are clustered or in rows, movable, for flexible grouping later in the lesson.

Students are discussing difficult problems from last night’s homework in small groups because the teacher recognizes that small-group work best meets the needs of the students in the class. Later she does whole-class and independent work to meet other students’ needs…

The teacher offers one student a second example of a math concept when the once given to the class doesn’t clarify the concept for him.

Students who are struggling with an assignment while a teacher is working with four students in the back of the room are working through a list of ‘What to do when I’m stuck and the teacher is not available’ ideas previously taught to them.

The teacher has two students who serve as ‘graduate assistants’ whom she knows have mastered the concepts and she has identified to the class as good resources if they have questions.

The teacher provides a few moments for students to think reflectively regarding the prompt before he guides their thinking. Those students how need intrapersonal contact appreciate the time to think, and many others who benefit from learning how to think reflectively.

These are all examples of teaching in a fair and developmentally appropriate manner – that is, differentiating instruction” (Wormeli 1-2)

Wormeli’s DI snapshots perfectly capture the strengths and weaknesses of DI. These teachers clearly know their students, adapt quickly to changes in the classroom and have thoughtfully prepared a variety of lesson types for their students. The methods of

differentiation are incredibly broad, though, and range from organizational – giving the student’s a list of strategies to get them started – to personal – giving the students with attention difficulties preferential seating. When every planned action of a teacher, though, becomes an aspect to differentiate, where is the heart of differentiation? Is differentiation central to the content acquisition or separate? Can every purposeful choice be considered differentiation? How do we disentangle types of differentiation and a successful DI environment from good teaching habits? Are they distinct entities? The category of differentiation is so broad that it may be helpful, at this point, to provide some categories.

At its most basic, DI allows for a diversity of pathways to for a student to access information – the Instruction piece of Differentiated Instruction. After instruction a student is measured to assess content acquisition, and, therefore, assessment can also be differentiated. An assessment that adequately measures the content acquisition of one student may not adequately assess the content acquisition of all students.

There are multiple starting points for differentiation. An educator can differentiate the way information is transmitted, the difficulty of the assignments the student receives, the type of assessment and more. An educator can also use DI to create a positive learning environment that is responsive to a student’s individual socioemotional needs, emotional and cognitive readiness and learning ability and disability (Tomlinson & Imbeau, 2010).

Best practices of DI start with a categorical approach which provides the context for learning:

1. Content- During differentiation, a teacher should emphasize methods as a

vehicle to acquire content specific knowledge.

2. Process- Content is accessed through activities that prioritize requiring

students to deeply interact with the material. Success in this area is monitored

through a series of formative assessments. Formative assessments are those

which mark content acquisition as a process with a series of check-ins rather

than only as an end product. The final assessment or cumulative product is

called a summative assessment. (Baron, 2016 pg. 47)

3. Product- assessments should be varied and allow students to show their

knowledge in both areas of specific content and the critical thinking process.

Tests and quizzes are the most common way of assessing knowledge, but they

are not, by any means, the only way of measuring content acquisition.

4. Affect- the motions and feelings of the student and the ways they impact the

learning process. Because the emotional state of the student is integral to the

education process, affect must be a part of the lesson and lesson planning

process. Affect can also include aspects of the students’ internal landscape

such as anxiety, learning disabilities, depression, attention disorders etc.

(Tomlinson & Imbeau, 2010 pg. 15-16).

Content is largely determined by curriculum which, in turn is determined by subject, grade, level, and school. At times is it federal or state mandated, while at other times teachers have the control to shape, without interference, the content of the course.

Process involves the acquisition of new content, which is found to be most effective when the teacher has a template lesson or strong examples, but is not heavily prescribed a lesson plan (Tomlinson & Strickland, 2005). Summative assessment strategies must allow all students of all learning styles, background and abilities to present the

information in a way that clearly demonstrates knowledge acquisition. Assessments must be designed carefully so as not to isolate students based on race, gender, ability and feelings of self-worth and value. Even the best intentioned of teachers can trigger hostile reactions from students if care is not taken to reduce and eliminate bias and stereotype threat in assessment (Good, 2017). Student affect is, perhaps the most critically impacted by the learning environment which is created by both the physical space and the teacher persona. Trust in the teacher, a welcoming atmosphere and a predictable environment can have huge impacts on the student affect in the classroom. Students with language-based learning disabilities, socioemotional considerations, attention disorders or autism spectrum disorders are especially sensitive to the classroom environment and teacher persona (Parker, 2017).

Since students can differ as learners in all dimensions, each must be accounted for as a teacher creates a fully differentiated lesson. Intrinsic differences such as background experience, culture, language fluency, gender, processing speed, learning ability and disability, socio-economic status etc. profoundly impact how a student learns most effectively, the type and nature of the scaffolding they will need and the points and which they will need the most and least scaffolding. Because these differences are individual to the student and changeable according the topic and content, the ideal approach to differentiated instruction is flexible and quickly responsive. This flexibility and individuality, however, must be balanced with the impracticality of writing and individuated curriculum for all students in the classroom. To combat this, students are often grouped according to ability or single lesson plans are aimed towards the middle ability students, disenfranchising both the students who excel and the students who

struggle. Differentiation is then often strangely paired with MI to inject interest and creative lesson plans into a static curriculum in the same way that learning styles and MI are often confused. With the best intentions, MI becomes a substitute for learning styles and a variety of modalities of learning become a substitute for foundational differentiation. In fact, the interplay between MI and DI is delicate, often tenuous, commonly misunderstood and not well balanced. When MI is understood properly, it can inform the creative decisions of educators. When it is poorly understood, it is mistaken for learning styles or ignored completely (Conrad, 2015). It cannot be overstated that MI can only inform the pathways of access or understanding, DI must provide the framework and access points while learning styles give insight only into the student preference. An activity conducted outside does not necessarily help a student who exhibits Natural

Intelligence learn an unrelated skill set and a lesson conducted outside may not intrinsically add value to a lesson. Differentiation does not center the student experience or add intrinsic value to the lesson has failed in its task, according the categorical interpretation of DI.

In this context, then, it seems that mathematics poses a particular problem. The content at a high school level starts moving towards abstraction, thereby making tactile and kinesthetic exploration more challenging. As mentioned before, meaningful instructional lesson templates are few and far between, often not age appropriate – they tend to be geared towards very young children – and sometimes sacrifice value for the fun factor. Assessment of mathematics has the same problem, it is either very traditional, or non-traditional in ways that do not meaningfully innovate the classroom. are not value additive. Students, during the high school phase, are often the most vulnerable to and

aware of the ways their moods and socioemotional landscapes influence performance, and often the least able to control the outcomes. So, with myriad of challenges, I wondered whether mathematics necessarily and categorically did not have access to the full spectrum of differentiation creativity.

Exploring Meaningful Differentiation in Mathematics

I refocused my attention. Was it possible to construct an entire math class that used differentiation as a central theme? Rather than differentiation being intricately tied with content, could I make differentiation itself the goal of the class? I became curious about explicitly teaching differentiation and pushing its relationship to the student experience to occupy a central role in the class, if necessary, deemphasizing specific content acquisition goals. I set a goal to identify key instructional and assessment differentiation techniques and then mold the lessons and content to fit the technique.

In order to focus the project, I started with three areas of DI. First, would differentiate content acquisition. Second, I would differentiate assessment, both formative and summative. Finally, I would differentiate for student affect, focusing on confidence and heightening interest through self-narrative and reflected identity. In each area, I drew out some core values and concepts before coalescing on the DI techniques that would take explicit centrality in the course.

When thinking about differentiation as it pertains to instruction and assessment, I wanted to shift the focus away from the teacher as the sole source of information and measurement of knowledge. Not only did I want the classroom the be student-centered, but also I wanted as many aspects of the class as possible to be student driven.

Generally, a teacher’s job is to select a pathway and then lead the student through the

material, assessing their competency along the way. Rather than dictate the flow of information, I wanted to act as a guide through student-selected material (Sankrithi, 2016 pg. 270). Integrating student interest and examples from the beginning of the class would allow the student to feel a part of, rather than apart from, the educational experience

(Dinkelman and Cavey 2015). Student generated examples may range from initial activity ideas and reference examples to exemplars of work and counterexamples. (Kinzel

& Cavey 2017). Beyond merely integrating student examples, I planned to encourage students to select integral units of study and then design meaningful activities around their topic of interest (Russo 2015-2016). Through active collaborative techniques cultivated as a class, they could deeply immerse themselves in the material (Ko, Yee,

Bleiler-Baxter & Boyle, 2016 pg 618-620). As the students become more practiced at designing their own topics of study, I hoped that active student involvement would inspire higher levels of experiential learning (Gates, Cordner, Kerins, Cuoco, Badertscher and Burrill 2016). Students would also take an active role in designing, grading and conducting the summative assessments by sharing their experiences and input (Rapke,

2017). Each assessment would require student input in the method of assessment and the rubric. After an assessment, student would be able, again, to take an active role in the grading process by completing self-assessments (Wormeli 2006, chapt. 4). In this way I wanted to foster student involvement and deep investment in the material (Watanabe &

Evans 2015).

To achieve these differentiation goals, I needed to recentralize literacy and language based critical thinking skills in the class. Students would be asked to verbalize understanding, analyze information, preview information sources and then read to find

meaning (Roepke and Gallagher 2015). After reading, students would use conversational strategies to encourage critical thinking to solidify the creation of links between the material both within the discipline and across disciplines (Keazer and Menon 2015-

2016).

Finally, I wanted to approach this class with a hyper-awareness of humanizing the abstract natures of mathematics. I needed to be aware and respectful of the way multicultural and socioeconomic backgrounds could impact learning and work to address those quickly and explicitly (Urbina-Lilback, 2016). For example, native language and home culture can have a huge impact on a student’s ability to process information, behave “acceptably” in the classroom and express their knowledge in a variety of ways.

Language fluency can impact the ability to acquire and express knowledge (Leith, Rose

& King, 2016) as well as have an impact on collaboration and sense of community with peers (Dong 2016). Language fluency may be impacted if the student is a non-native

English speaker, is multilingual or has a processing, speaking or learning disability.

Beyond cultural and ability considerations, I also wanted to be aware of the ways that interpersonal skills or lack thereof can impact participation and clarity of presentation. I also needed to be exceedingly aware of the role that anxiety can play in performance or the ability to express clear mathematical thought (Wilson 2013).

Since each student comes to class with a unique story, history and identity, I wanted to use the student experience to create comfort in the classroom by linking identity to the history of mathematics. Clear discussions of race, gender, ethnicity, and sexuality will challenge some students and address the integral identity politics of others

(Rubel 2016).

Finally, I wanted to use technology in the classroom in a way that highlighted learning opportunities and added a clear value to the experience. Technological integration is often touted as an easy way to differentiate for readiness and content remediation. After examining a plethora of programs, including Khan Academy and other online teaching platforms, I opted not to differentiate for technology specifically. If a specific application, website or program presented itself that would add to the lesson in a content-value additive way, I would include it, but I would not specifically seek out technological differentiation.

Using student input, centralizing literacy skills and humanizing mathematics, I designed a new approach to a traditional curriculum.

Chapter III

Course Design and Execution

Overview

I teach at a small independent high school in the greater Boston area. Classes are purposefully kept small to better give the students individual attention and encourage a close teacher-student relationship. The school’s mission statement embraces both the theory of Multiple Intelligences and the ideals of Differentiated Instruction. Math classes follow a common trajectory (see Figure 1), with more choices available as electives at the higher levels. This class, titled Survey of Advanced Topics in Mathematics: History,

Philosophy and Art, is designed as a Senior elective, available as an option to Juniors who have completed Algebra 2, a graduation requirement, and Sophomores with departmental permission who are taking this class concurrently with another math class.

The class was initially proposed to the curriculum and described to the students with the following text:

“The study of mathematics in an ancient discipline, almost 4000 years old. Records of mathematical thinking, though, date back almost 11,000 years. From numerics to algebra, game theory to engineering, geometry to calculus, mathematics is a wide and varied field that encompasses many areas.

Survey of Advanced Topics in Mathematics is a class for Seniors and Juniors who have completed Algebra II. In this class, we will examine various topics in mathematics through historical, biographical and artistic lenses. Starting with the history of numbers, the class will explore the creation of mathematics in both practical and philosophical sense. We will progress through logic and early geometry before moving into the realms of game theory (including game creation), set theory (including problems of infinity), combinatorics (including probability and advanced counting techniques), cryptology (making and breaking codes), higher dimension geometry (including curved space and 4+ dimension calculation) and historically important and potentially unsolved math

problems (including squaring the circle, fractals, Goldbach Conjecture and Fermat’s Last Theorem).

Students will use a variety of media in their learning including an examination of Math in Pop Culture (The Mathematics of the Simpsons, the TV Show Numbers, the movies Proof, A Beautiful Mind and Pi) and Music (harmonies, melodies and frequencies). We will also use artistic platforms to express mathematical ideas (creating fractals, tessellations, perspective etc.) and finding the mathematics within artistic movements (Fibonacci sequences, the Golden Ratio, Symmetry and Beauty). Finally, we will leverage storytelling and social history to discuss the people and personalities behind the mathematics (The Calculus Wars between Newton and Leibniz, the Pythagoreans, Ada Lovelace and Alan Turing, the Misunderstood Cantor and the Impossible Gauss).”

As the focus and goals of the class changed, so did the description in the syllabus. The changes reflect the concentration on student lead topics of study and the concentration on creativity and differentiated instruction. For students, the goal of this class is to gain an appreciation of the scope of mathematics and the discipline to explore high level mathematics in a multi-disciplinary and fun manner.

It is designed to be a Survey of Topics with an emphasis on the historical context of the topics, an exploration of the personalities of math and in-depth discussions of the theories and applications of the topics. The class is purposefully high conceptual understanding with low emphasis on calculation or advanced mathematical knowledge fluency. It is designed with an agile framework to best address student interest in topics and pace of learning. It is topic oriented, as opposed to temporally linear (like a traditional History class) or sequential (like a traditional Mathematics class). The topic- oriented nature is designed to maximize student interest and buy-in to the material.

Figure 1. Scope and Sequence of Mathematics Classes

Students are required to take through Algebra 2 and most are encouraged to take a class, PreCalculus or above, their Senior year.

After the initial core-value search of DI concepts, I picked out several observations that inform day-to-day class design. First, Differentiated Instruction is most effective when the topic is revisited multiple times with multiple instructional techniques.

Second, DI is most effective when it is paired with a metacognitive element. Third, DI requires student input to be most effective. Finally, students are not always aware of their own best practices and therefore their perceived comfort zone may not always yield the best results in class work or on assessments. The qualitative data gathered from the students will address several topics key to differentiated instruction. Surveys will ask about preferences in instructional techniques, assessment preferences and equity, how confidence influences performance and general attitude towards mathematics. Full text of all survey used in the class may be found in Appendix 5. Information about grading strategies and student feedback and self-evaluation will also be included. The qualitative

data will inform the sequence and scope of the class and may influence the strategies of instruction and assessment. Students are familiar with metacognitive frameworks and habits of mind from the general school philosophy. Data will generally be aggregated unless it is imperative that I be able to tie the identity of a student with the response.

Timeline of Class

The year is divided into trimesters. Each trimester, one topic of study acts as an anchor unit. This is a topic foundational to the understanding of mathematics from a historical or calculation-based perspective. Students will then be given a list of topics to choose from or be asked to formulate their own ideas for new topics of study. Topics will be chosen by majority consensus, although students will be allowed time to persuade their classmates of one topic over another. For the initial list of anchor topics and alternate possibilities, see Table 1. Topics that are not chosen will be offered as options later in the year or be encouraged as topics of individual exploration for independent study. Topics that are perpetually unpopular will be removed from the list or rephrased to promote interest. Students also have the opportunity to generate new topics as their interest changes or as an overwhelming interest in a topic takes places. Students will be given ample time over the course of the year to independently research topics. For a list of student generated topics from 2016-2018, please see Table 2. Each unit of student will take three to six weeks depending on the complexity of the topic and the depth of exploration.

Table 1. Topics for Consideration

Anchor Topics Trimester 1 Number Systems and History of Numbers

Trimester 2 Art and Beauty (if this was chosen in Trimester 1, it will be substituted with Music and Mathematics)

Trimester 3 Finances and Money Math (If this was chosen in Trimester 1 or 2, it will be substituted with Epidemiology and Statistics)

Additional The Problem of the Primes Units of Study

From Counting Sticks to Smartphones: A History of Computers

More than 3: A Discussion of Dimensions

How to Lie with Statistics

Optical Illusions

Game Theory: An Introduction

Logic and Fallacy

An overview of anchor topics and optional units of study offered to the students.

Table 2. Student Generated Ideas for Topics of Study 2016-2017

2016-2017 Math of Betting and Gambling

Mathematics of Ecology and the Environment

Mathematics and Sports

The Opioid Epidemic or Gun Violence

2017-2018 Bit Coin and Cryptocurrency

Ballistics

Math in the Simpsons, Futurama

Forensics/Ethics

Meteorology and Predictions

A sample of student generated topics from 2016-2018. Not every topic was adopted as formal topic of class study, but each student was given the opportunity to explore areas of personal interest in depth.

Each topic will start with an exploratory exercise. These exercises be open- ended, exploratory, be done individually or in small groups and act as a general introduction to the topic. The lack of prescriptive structure will allow students to enter the topic regardless of readiness level and allow the student to find interest and challenge at an appropriate level.

Information acquisition will start with definition of terms, historical reference, context and relevance and a review of key mathematical concepts. Introductory notes will be given in a variety of formats depending on the complexity of the information. Notes may be highly templated or free form. Students may be asked to interact with the information without taking notes or may be asked to take notes from a text or other print resource.

Information exploration will take place with a series of short in-class games, activities or mini-projects. These will allow the students to explore the information in a variety of ways, accessing information at the level and format that is appropriate for the individual. At times, the students will be allowed to operate within their comfort zones and at other times they will be required to perform the task in a specific way. This will allow students who demonstrate strengths in the prescribed method to work at ease and other students to stretch their comfort zone. Because these methods will change over the course of the unit, no student will be required to consistently work either inside or outside of a comfort zone.

Assessment will be discussed as a class and may take several forms ranging from traditional written test to project-based. The school requires two of three trimesters to culminate in a written exam, one of which must be a formal final exam with a three-hour

testing period. Summative assessments during the trimester may stand alone (one culminating assessment of knowledge acquisition) or may be paired (for example test and project) to maximize the types and qualities of information presented. At times the students will be able to choose their method of assessment and at times the method will be chosen for them.

After all summative assessments, students will be asked to engage in metacognitive activities that encourage them to reflect on their study habits, confidence levels, preparedness, interest and engagement with the topic, efficacy of academic habits etc. Records of these responses will be given to the students and periodically they will be asked to revisit their metacognitive reflections.

After the cycle of the unit, students will be given the opportunity to choose the next topic, or I will insert the anchor topic and the cycle will begin again. We started the year with the anchor unit: Number Systems and the History of Numbers.

Chapter IV

Sample Lessons with Discussion

The following lesson were conducting between September 2017 and December

2017. Some of the lessons had been taught the previous year and some were new. These lessons do not represent the entirety of the unit, but, rather, have been chosen because of the range of DI techniques they employ. I have presented the lesson plans with all accompanying resources. After each lesson, I will briefly discuss my observations about the class as a learning entity and any observations of the lesson itself.

Lesson 1: Information Acquisition

Differentiation Type Information Acquisition

Method of Differentiation Students will be given template notes and will be able to

work through the information at their own speed. I will

read aloud and walk through the first example on a white

board for those students who need it. Students may

follow along or work at their own pace. Different areas

of the room will have different work expectations. One

group of tables for students who want to work with me, a

table for those who wish to work with partners and

several spots around the room for those who wish to

work independently.

Differentiation within the Students who need additional instruction or individual task attention will be able to access me during class. Students

will be allowed to work with a partner, although not

explicitly encouraged to do so. Students who need a

challenge will be given additional concepts to explore.

Any question labelled (Challenge) will be heavily

encouraged for students who acquire the baselines

knowledge quickly and/or easily.

Unit Number Systems and the History of Numbers

Subject of Notes Changing Bases

Prior Knowledge Students at this point have a clear concept of the

difference between a positional and an additive system.

This is an introduction to bases within a number system

and the methods needed to change between number bases

Confidence and Mood Prior to the introduction of this concept, students were

Evaluation asked to measure their confidence about the last topic

and their excitement about learning a new concept. The

majority of the class replied in the positive/affirmative

Additional Resources See Appendix 1 for a reproduction of the template notes

and worksheet.

In other classes, I rarely work with highly templated or guided notes. Although, as is shown in Figure 2, students did not choose template notes as an area of strong preference, I wanted to experiment with the form for self-guided instruction. For those students who needed little or no teacher intervention and for those students who prefer to discover information without an intermediary, this was an incredibly effective lesson.

They were able to work at their own pace, look up information as they needed and challenge themselves as was appropriate. I did not explicitly encourage students to work together, but those that did were able to use collaboration to check their work and challenge themselves.

For those students, though, who struggle with calculation-based thought, have high anxiety surrounding math or need a greater amount of teacher intervention, it was unclear how much of a difference the highly guided notes made. Each of the students felt more comfortable and confident with a guided conversation or one-on-one interaction.

Some students seemed overwhelmed by the amount of information on the page. Students who self-identified as needing a challenge felt appropriately challenged and some used the opportunity to teach peers the methods of calculation.

Given the chance to repeat this lesson, I would change only a few aspects. First, I would give a shortened template so we, as a class, could work through one problem together. Then, I would give the highly guided template to the class, allow the independent workers to differentiate themselves and guide self-selecting students through one or two more examples. Second, I would split the lesson across two class periods and offer more opportunities for practices. Switching from changing to base 10 to from base

10 very quickly proved to be more time costly and confusing than necessary. Finally, I

would modify my confidence and mood evaluation. I asked students about their comfort level with the previous topic and if they were ready to learn a new topic but did not account for the complexity of that topic. Instead, I would introduce a simple or familiar base change idea, converting hours to minutes, for example, or centimeters to meters, gauge their initial understanding and then ask about conceptual readiness. Asking for confidence levels and readiness without a context does not necessarily give valuable information.

Figure 2. Differentiated Instruction for Content Acquisition.

Student response to the question: What type of notes do you prefer?

Lesson 2: Summative Assessment

Differentiation Type Summative Assessment

Method of Differentiation As a summative assessment, students will be asked to do

both of the following tasks:

• Take a traditional test

• Do a project

Students will be asked to choose their weighted percent

for the test/project. By choosing their percentage split,

students will have the opportunity to be graded on

perceived strengths and weaknesses or self-identified

learning modalities. Asking to choose a weight will also

allow the student to build realistic time expectations and

executive functioning skills.

To encourage honest effort on both the test and the

project, I mandated that the weight of a single task could

go no higher than 80% (or no lower than 20%). A “true”

or traditional split for the student was 50/50. For a list of

classroom splits over the last two years, see Table 3.

Differentiation within the Students would be allowed to self-differentiate in both task the test and the project.

Test: In most sections, students had more choices of

questions than they were required to answer. Some

questions were marked as “Challenge” questions to allow

students to self-differentiate to a higher level.

Project: Students could select the level of complexity

appropriate for them. Indicators of challenge were put in

the project checklist and discussed with each student

individually.

Unit Number Systems and the History of Numbers

Timeline of Assessment This assessment was designed to be completed over the

course of four class periods:

Day One: Introduce Project and idea of different

weighted averages. Have the students submit an initial

idea of the weights of the grade.

Day Two: In-Class students brainstorm “What is going to

be on the test?” by looking through their notes and then

coming to the board to write an idea or concept. A

teacher can use this as a class participation grade or a

formative assessment grade. Tell the students to write

their initials next to their suggestions and then, after the

activity, take pictures of the board. This strategy can

sometimes kick start a lethargic class and give students

with anxiety disorders diffuse instead of directed

attention (Parker 2017). Spend some time in class

reviewing what are KEY topics and what are DETAIL

topics. Give the Study Guide, in Appendix 2, and the

start to conference with each student to determine a final

decision on the test/project split and clarify any questions

about either portion of the assessment.

Day 3: In class work day. Students may work on project

or study guide.

Day 4: Projects are due, and test occurs. See Appendix 2

for text of test, study guide and project expectations.

This style of assessment is a favorite among my students. They feel it allows them to best represent their knowledge while covering all pertinent information.

Additionally, it allows for the students to personalize the assessment in order to account for factors outside of the classroom. Students reported feeling more ownership over their grade, their investment and the material. The latter surprised me slightly because the expectations of content competency did not change but the students felt more invested in the process.

If I were to alter the process slightly, I would ask the students about their rationale for choosing the percentage split. Are they operating within their comfort zone? Are they unsure, so choosing a traditional split? Are they hoping to challenge themselves and so choosing a more disparate split? Eliciting this information from the students will inform me about their confidence levels, their preferred assessment styles, their willingness to take risks and the areas that they may need to be encouraged to explore in the future.

Table 3. Breakdown of Student Selection for weighted test/project

Year Split Number of Students Test/Project

2016-2017 20/80 4

30/70 1

40/60 1

50/50 3

60/40 1

2017-2018 20/80 4

30/70 2

40/60 1

45/55 1

50/50 5

60/40 1

Students chose their split after the topic was introduced. They had to submit their proposals in writing. Then, I had a small conference with each student to get verbal confirmation of their choices and allow them to change their mind. After this point in time, students were not allowed to change their elections.

Transition Between Units

After the summative assessment, students were given the option to share their projects (oral presentation). No students took the offer, which may indicate a lack of confidence (mood) or a general dislike of presentational demonstration (assessment style preference). Oral presentation has been noted, see Appendix 6, as a hesitation or weak point for the class. To address this, I will specifically address it as a differentiated focus later in the year. Students were then asked to reflect on their project and tell me what went well and what did not go well. They could email this to me or write it on paper and turn it in. Finally, students were asked to fill out a Google Form to help select the next unit of study. I generated the first list of options, briefly verbally explained what each topic would entail and allowed a brief amount of time for internet research into the topics.

I included an option for students to write in requests that did not appear on the list. These requests, see Figures 4 and 5, were then integrated into the options for subsequent units.

The results, found in Figures 3, 4 and 5, were not tallied live to add a feeling of anticipation and excitement for the students. When students entered class the next day, each was given a secret message in the form of a simple acrostic, announcing the new unit.

Figure 3: Student Selection of Course of Study – First Choice

Students were asked to select their first, second and third choices for the next topic of study. Winner was chosen by majority consensus.

Figure 4: Student Selection of Course of Study – Second Choice

Written-in choices (anything appearing after the first four option) were taken into consideration for the next round of student selection of direction of course.

Figure 5: Student Selection of Course of Study – Third Choice

Popular second or third choice responses were kept on subsequent surveys. Written-in student responses were included in subsequent surveys.

Cryptology and Cryptography is a particularly complex unit that spans thousands of years of history, multiple cultures, several intricate mathematical techniques and has a significant amount of possibilities for technological integration. With the new focus on

DI methodology and the time needed to focus on metacognitive work, student centered rubric planning and grading, etc., I anticipated that this unit would last approximately five weeks. Classes meet three times a week for a total of three hours and forty-five minutes of classroom experiential time meaning that students had a total of 18 hours and 45 minutes exploring this topic of study. Of that, approximately one third was dedicated to student-generated activities, course creation and independent study work time.

Lesson 3: Formative Assessment

Differentiation Type Formative Assessment

As a Formative Assessment, students will be asked to:

1. Research a topic

2. Design an informative poster

3. Explain their research in presentations to their

peers and teacher.

Method of Differentiation Students will be asked prior to the project two

questions. First, they will be asked about their

perceived understanding of the material using a 3-value

– great, okay or neutral and not very well. Then they

will be asked to identify students who they would like to

work with in a group setting and students who they

would prefer to avoid. Students will be grouped

according to confidence level, although not necessarily

in homogenous group, and to mitigate interpersonal

conflict. Students will not necessarily be group

according to request. Complexity of the topics will be

given based on confidence level and group dynamic, see

Table 4 details.

Differentiation within the Students will be able to select which roles they would task like within the group.

Unit Cryptology and Cryptography

Timeline of Assessment This assessment is designed to take place over the

course of three class days.

Day 1: Students are given assignment, see Appendix 3,

and groups. A teacher may either assign topics

(teacher-initiated differentiation) or allow students to

select the topics (student-initiated differentiation).

Students spend a small amount of time researching their

topics. Research is assigned as homework.

Day 2: In-class work day. Students may design their

posters, assign roles, and/or practice their presentations

and explanations. Students may be allowed to choose

their roles of presenter or scholar (as in my class) or the

teacher may assign roles to stretch students’ comfort

zone or allow students to operate within a comfort zone.

Day 3: Allow students15 minutes at the beginning of the

class to finish last minute details and select a spot to

hang their informative posters before the activity begins.

Homework- Students will complete a reflection about

their role in the project, what went well, what did not go

well, etc. For an example text of student reflection, see

Appendix 5.

Activity Details Gallery activity is meant to:

• Encourage physical movement around a

space

• Allow students to interact with information

in an auditory and visual way

• Elicit good note-taking skills

• Build confidence in student ability to convey

knowledge and expertise

Experts will start at their poster and stay there for the

remainder of the activity. Scholars will start at any

other poster they choose or start at an assigned spot to

ensure even distribution and increase focus for easily

distractible classes. Teacher will set an alarm for 5

minutes and the activity will begin.

In those 5 minutes, the Expert will present the topics to

the Scholars. The Scholars are expected to ask

questions, take pictures, participate in activities, and

collect notes. When the alarm sounds, Scholars will

shift to a new poster and begin the process again. This

process repeats until each Scholar has visited each

poster. Flow of students can be highly moderated, as

mine was, or more freeform as is appropriate for the

class. I assigned both starting positions and a direction

of movement.

Once a Scholar has visited each station, they will return

to their Expert and relay the information they acquired.

Experts are expected to copy or collect the notes and ask

clarifying questions. This process should take 15 to 20

minutes. Finally, students will be able to gather in a

single group to ask any final or clarifying questions.

As a teacher, I visited each group and took notes on:

• Completeness and accuracy of information

presented on the poster

• Ability of Experts to verbally explain the

material and answer questions

• Interactions between Experts and Scholars

• Quality of complied notes and questions

asked by the Scholars

This activity was particularly successful as a teaching and learning tool and feedback from the students on the self-evaluations was overwhelmingly positive This assignment was particularly well differentiated as it allowed for both teacher-initiated and

student-initiated differentiation. The multitude of options available to the students and pace of the timeline allowed each student to demonstrate their knowledge in their preferred method and to the best of their abilities. I was particularly impressed, though, at the number of students who stepped outside of their comfort zone, choosing to be an

“Expert” even though they stated that oral presentation was not their preferred method of assessment, see Appendix 6.

Due to the positive feedback and quality of the content acquisition, I will be using this activity again. I would like to experiment with the idea of using it as a summative assessment or culminating activity for a unit. Before using it as a major assessment, I would discuss with the students what makes a good presentation. In this way, I would be able to also incorporate student input about grading a presentation.

The biggest piece of critical feedback from the student self-evaluations was that the notes from the Scholars were not clear or not complete. I can see this being an excellent activity to teach note taking and communication skills or further scaffolding to allow for students who have different levels of organizational competency.

Table 4. Group Work Assignments and Difficulty Levels.

Topic Complexity

Polybius Square Expected

Semaphore Expected/Low

Morse Code Expected/Low

Vigenere Cipher Challenge

Playfaire Cipher Challenge

For this project, I assigned topics based on previous class performance and self-reported confidence levels with the material.

Lesson 4: Information Acquisition

Differentiation Type Information Acquisition

Method of Differentiation Students will initially conduct their own research

about a topic. Then, collectively, the class will

compile notes

Differentiation within the Students will research and share knowledge at their task preferred challenge level. Students may be given

guiding questions or specific resources to use.

Resources may be labeled as Challenge if the

language is particularly technical or high-level.

Simpler texts may be given to students as needed.

Unit Cryptology and Cryptography

Assignment Assigned as homework: Students were told to learn

about the mathematician Alan Turing. They must use:

• At least 3 reputable websites

• At least 1 print source

Students were told to find information in three areas:

1. Turing – The Man (biography)

2. Turing – The Machine (breaking Enigma)

3. Turing – The Mathematics (other major

contributions to the field of mathematics)

Students need at least 10 facts and at least 2 facts in

each category.

Assignment can be altered in the following ways,

depending on classroom need.

1. Students can be assigned a single category or

all categories.

2. Students can be required to formally cite

sources or merely acknowledge their sources.

3. Students may be given a template, see

Appendix 4, to assist in research organization.

Activity Before Class: Teacher sets up three distinct zones in

the classroom labeled with the titles of the three areas

of research. These areas may be white boards, desks

in clusters with posters sticky notes on walls etc. as

the classroom can manage.

In Class: Students will spend 10- 15 minutes writing

down their facts in the appropriate places. After the

fact gathering has occurred, students will be assigned

to each category and asked to edit the information for

20-25 minutes. They are tasked with the following:

1. Arrange the information either chronologically

or thematically

2. Eliminate duplicate information

3. Fill in any gaps in the information. They may

use internet or available print sources.

Finally, students will then be asked to share or

summarize their information with the rest of the class.

Students who are not presenting will be encouraged to

take notes, take photographs of the boards or posters

and ask questions to the presenting groups.

Differentiation Goals The activity is meant to:

1. Encourage movement around a physical

space

2. Encourage collaboration and

communication in the fact-editing process

3. Identify key information and eliminate

extraneous information

4. Present the information graphically and

verbally

5. Collect information auditorily and through

writing

6. Allow students to build confidence and

ownership over the information presented.

7. Introduce a historical figure to connect the

mathematics and concepts to history and

biography.

Teacher Specific After the information gathering phase, I ask each

Information (methods of student to turn in their fact sheet. During the editing

grading and further process, I take pictures of each group’s work. As the differentiation) groups present, I take notes on accuracy and

completion of task. In this way I can assess level

participation in the activity and involvement and effort

in the organization and presentation.

For further differentiation, teachers may assign roles

within the groups. Roles may include:

• Chief researcher

• Information organizer

• Information recorder

• Information presenter

Teachers may allow students to self-select roles,

assign roles within a student comfort zone or assign a

role meant to stretch a comfort zone. For this activity,

I allowed students to self-select roles but explicitly

encouraged them to step outside of their comfort zone.

More than half of the class chose a role that was

outside of their self-reported comfort zone.

This is a particularly central activity to differentiate content acquisition. It is completely student driven and places the students at the center of the information acquisition process. At every stage, students collect, curate and control the access to

information. It is also crucial as a lesson that focuses predominately on the human aspect of mathematics and allows students to link mathematical concepts both within the discipline and across disciplines. Through discussions of Alan Turing, I am able to explicitly address questions of access and equity in the discipline of mathematics.

Alan Turing’s work with Enigma represents the first time that modern computing was used in cryptography. He is now credited with creating the framework that allowed for the invention of the personal computer. Second, studying Alan Turing can be a valuable entry point to discussion of modern issues and ethical quandaries within mathematics. Much of his mathematical discussions focused on Artificial Intelligence and the nature of consciousness. Asking students to think critically about the “Turing

Test” and examine larger questions about consciousness and responsible use of AI technology can help create links from the past to the present and encourage critical thought about modern issues. Finally, discussions of Alan Turing lead to discussions of diversity in the field of mathematics. As a homosexual, Alan Turing was denigrated, denied praise for his innovations, purged from public records, criminalized and driven to committing suicide. By explicitly including him in discussions of mathematics, we can continue to deconstruct questions of access and identity in the field (Wamstead, 2017).

My class was particularly fascinated by questions of Artificial Intelligence and the

Nature of Consciousness. We watched several clips of the movie “Imitation Game” and then I devised an additional activity. Because of the agile and responsive nature of the design of the class, I could follow student interest rather than adhere to a strict sequence.

The students and I designed a classroom activity to examine more directly, the Turing

Test.

In this activity, two students were asked to leave the room. The remainder of the students stayed in the room and were either questioners or judges. In my classroom, I had three enthusiastic questioners and the remainder of the class was “judge”. The questioners needed to formulate a question, write it down on a notecard and slide it under the door. One of the students in the hallway needed to answer the question with their own thoughts. The other student could use only Ciri or other predictive text program.

Both answers were returned to the classroom and the judges were asked which answer came from a human and which answer came from artificial intelligence. After two or three examples, students reconvened to share their observations. We ran this experiment twice, with different students as judges, questioners, humans and AI.

After the discussion, which was conducted as a whole class, each student was asked to write down their thoughts, observations and feelings about the activity as an Exit

Ticket.

This impromptu activity, inspired by student interest garnered by the DI information gathering assignment was successful because of the agility of the lesson planning. Allowing the time to add an additional activity provoked deep though from the students and let to formative academic discussions. Student felt ownership over the material, challenged at the appropriate level and were given the space to foster their own interest in the topic. Additionally, in my diverse classroom, we were able to explore concepts of access and equity. Those students who were able to “see themselves” in Alan

Turing left with a feeling of empowerment. Those who could not immediately identify with the mathematician were given the sense that mathematics, a largely abstract field, came from a human element. The success of this activity has prompted me to choose at

least one mathematician from each chosen topic of study to explore on a more personal level. Since then, we have spoken about John Nash (Game Theory and his struggles with mental health), Katherine Johnson (Astronomy and her struggles with sexism and racism), Euclid (Geometry and the whitewashing of mathematics) and Ada Lovelace and

Grace Hopper (Computer Science and gender bias within mathematics). An in-depth discussion of each of these figures allows students of all backgrounds to see themselves within the field of study and acknowledge the humanity in an often solely abstract field of study.

Conclusion

These lessons were conducted over the course of two months in my classroom. They are not every lesson conducted between September 2017 and December 2017, but they are representative of all lessons. Lessons begin with a strong foundation in Differentiated

Instruction, center the student experience and allow for both teacher- and student-initiated differentiation. Each lesson use an MI framework as creative inspiration and then allows the students to access information or demonstrate knowledge using a variety of learning styles and preferences. Lessons actively seek to connect the students with the human nature of mathematics and explicitly address issues of access and equity within the classroom.

Chapter V

Reflection and Analysis

For a period of eight weeks, I closely scrutinizing the class and course design, catalogued student responses, varied instruction to match student self-reported preferences and interests and actively and explicitly centered the student experience. The approach to instruction and assessment was carefully recorded with an eye towards making this particular class and its approach a permanent staple at the school. After the completion of the formal process, I began to wonder how many of the lessons, ideals and approaches were unique to my school and situation and how many could be generalizable. What follows is a reflection of the class as a whole and major takeaways for other teachers.

Focus on Differentiated Instruction in the Classroom

By shifting the focus from the content of the lesson to the methods of differentiation, I was forced to more deeply examine the content and my dependence on the content to dictate the differentiation of the class. I was made aware of how central a role content often fills and how, at times, best instructional methods are sacrificed for required content acquisition.

My original project included differentiating instruction, differentiating assessment, and differentiating according to the socioemotional needs of the student.

Focus on the first two, with an emphasis on differing learning styles and inspired by a multiple intelligence framework, allowed me to create more substantive, more creative

and more level appropriate lessons. Asking myself how I was teaching rather than what I was teaching refocused my attention on student need.

Differentiating assessments encouraged me to reexamine what was crucial knowledge and how to elicit demonstrations of knowledge that were integrated and foundational to the topic. Building assessments that measured strengths, insights and understanding is more complicated, but in the end better reflects the actual understanding of the student. More and more, I found myself asking not if I had effectively taught the students content, but how to design a system that would allow the students to demonstrate what they had learned. At times, the answer was uninspiring. The student could not adequately demonstrate what they had learned or that they had learned. Over the course of the class, though, I found the students better able to demonstrate clear understanding and, more surprisingly, demonstrate more complex and ambitious levels of knowledge

(Booth 2003 pg. 291-294). Although each student had a preferred method of assessment, see Appendix 6, Figure 6, as the year progressed, students were more likely to reach outside of their preferences and choose projects that best fit the material, rather than their interests.

I am not sure that a focus on differentiation alone could have provided this shift in attention for the students. As a way of gathering information about the success of the differentiation, I included student input at every level of the course. Students were asked about new topics of study (See Figures 3-5 and Tables 1 and 2), confidence levels (see

Appendix 6; Questions about confidence and concern), preferred method of assessment

(See Appendix 6, Figure 6), content acquisition and instructional preferences (See

Appendix 6 and Figures 2 and 7) and much more. Students were asked input in the design

of project rubrics (See Appendix 6), weighted percentages of exams (See Table 3), and presentation expectations. I had originally intended the surveys as a way of gathering information for this project and my own, teacher-centered interaction with the class. The surveys, though, developed into a major driver of the success of the class and students.

Allowing the students to have input at every level, ensured that students were more likely to engage, explore and feel ownership over the material. By asking the students what they wanted to learn, they started to become active creators of their own knowledge rather than passive recipients of abstract concepts. By asking the students how they preferred to acquire information, they were more likely to experiment with methods of differentiated instruction rather than rebel against notes or wait to receive teacher-led lessons. I also found that students who had strong preferences (templated notes vs. notes from the board, for example) started reaching outside of their perceived process. Some discovered that their area of preference was not necessarily an area of strength.

Interestingly, by drawing attention to the differentiation, students were more likely to experiment and be self-aware of their own needs. I was initially concerned that the focus on differentiation, especially when it came to content acquisition would encourage students to stay inside their comfort zone. The opposite appears to be true. The metacognitive process changed the process.

Similar trends appeared when I asked the students about assessments. Asking the students how they preferred to demonstrate their knowledge made them a part of the process, forced them to look beyond what I “wanted to see on a test” and into how they could show me what they had learned. In those circumstances where students were allowed to choose any method of assessment (project, presentation, essay, test, etc.),

students were as likely to choose inside of their comfort zone as outside. On several occasions, students mentioned that they chose a style because it wasn’t their preference and they wanted to see if they could succeed. In cases where I allowed the students to weight their tests/projects I found that initially students chose within their comfort zone.

Students who self-identified as “bad test-takers” chose weights that favored projects, students who wanted a true reading of their scores chose even weights and students who preferred a formal test chose the appropriate weights. As I recreated this option though, students were more willing change their minds and experiment. Some students would mention that they wanted to hold themselves accountable and so chose more even weights. Other students selected a larger mismatch out of their comfort zone to give themselves a different challenge.

Additionally, involving students in the creation of rubrics not only made my expectations as a teacher concrete and apparent, but also allowed them to incorporate their own expectations (See Appendix 6). In making the expectations explicit, each student was able to understand, apply and embody the expectations. Making expectations and requirements explicit, though, was less impactful that making the process open, integrated with most lessons and obvious to the student. Again, the metacognitive process seems to have changed the process.

Differentiating for a student’s socioemotional needs was, perhaps the most impactful and least measurable. My biggest shift in the classroom was to humanize the mathematics giving it a face, a historical context and a relevance to the lives of the students. Traditionally marginalized groups are represented at much lower rates in

STEM fields and high-level mathematics classroom. Historical assumptions and

stereotype threat are two of the many reasons for this lack of representation. I sought to actively counteract the cultural narrative that mathematics was abstract and impersonal, historically dominated by white European men and was somehow removed from interdisciplinary and creative work. I included activities that purposefully challenged the notions of straight male Eurocentricity and then encouraged seemingly off-topic academic conversations about the mathematicians or historical contexts. Allowing the students to choose the path of study allowed the individuals to see themselves in the mathematics. Encouraging conversation elicited in-depth thought from some traditionally introverted students. It allowed the students of color in class to discover role models in the field of mathematics. It allowed the non-American students to share their home history and culture with the class. It challenged the white male students to question the current face of mathematics. It allowed the student who self-identified as “not math people” to integrate areas of comfort like verbal, written or artistic expression into a mathematics class.

Conversational techniques helped build a common vocabulary around equality, equity and prejudice in a field that does not usually invite these conversations. A particularly interesting conversation followed a class viewing of the movie “Hidden

Figures”, a film chronicling the lives and work of Katherine Johnson, Dorothy Vaughan and Mary Jackson, three African American women, mathematicians, computer scientists and engineers, who worked with the NASA space program Project Mercury in the 1960s.

Several students left the movie stunned, asking why they had never learned about these women in school. Setting aside the irony of this question being asked in the classroom, I was struck by their reaction. Students who had studied math for their entire school

career, including some of the calculations used by the characters in the film, had never been introduced the humans who had pioneered the field. For the students, mathematics was not a human venture, but an abstract series of rules. Some of my female students of color had never seen a professional black woman do math. In the classroom, there were no role models. There was no avenue in school for students to connect their personal lives to the material. One student mentioned that she had never thought of herself as a mathematician because math was for white folks. Even though this sentiment might not be consciously taught in the classroom, the lack of representation in the STEM fields and the lack of explicit conversations about race, sex and identity in the classroom leads many to the idea that mathematicians are not like them. By allowing students the space to humanize the mathematics, the students were more likely to engage in the mathematics themselves. Inviting non-mathematical conversation opened avenues for mathematical exploration.

In addition to explicitly exploring ideas of identity in mathematics, I found the unambiguous discussion about the role of self-confidence and self-advocacy in the classroom to have an enormous effect. Issues of confidence, especially among those students who feel disenfranchised about the field of mathematics, effect performance in the class and the likelihood of a student to pursue mathematical studies in the future.

Involving the role of confidence publicly and then working, as a class, to build awareness and skills to boost confidence both destigmatized the academic impact of anxiety on assessment and performance and allowed the students to use each other as resources, rather than rely on themselves or their teacher for security. Students learned to trust their

peers, ask for help without being self-conscious and recognize that their inherent abilities and skills were not necessarily tied to a single performance.

The emphasis on differentiating instruction seems to have altered my approach to both the content and the teaching process, by design. By relinquishing the centrality of the information in the lesson, I was able to give the students more of an ownership of their own content acquisition. By involving the students explicitly in every aspect of the class, the students felt ownership of the course and were more willing and more able to perform at high levels both in and out of their comfort zone. Surprisingly, though, the most transformative aspect of the class was not the differentiation itself, but the involvement of the students in the creation of the course, the explicit conversations that surrounded instruction, assessment and grading and the emphasis on metacognitive processes. Students felt ownership over the class, understood the reasons for assessments, were able to demonstrate knowledge within a context and were able to explore their own though processes with full agency choose their preferences while feeling the confidence to try new instructional or assessment styles.

The ideas of integrating metacognitive exercises in a class are not new.

Particularly with the current educational focus on Growth Mindset, many classes and teachers across the nation are recognizing the need for students not only to learn the material in the classroom, but also to focus on the mental processes that allow for content acquisition. In my own classroom, framing each lesson with a metacognitive component and then asking the students to examine their own thoughts, styles, preferences and identities in the context of the content, expanded the possibilities for the students. Over time, students were able to make clearer connections within the discipline and across

disciplines. Students were also, anecdotally, more invested in this class than in their previous math classes because they felt they were able to express their individuality and personality through the mathematics rather than being asked to subsume their interests and intrinsic abilities for the sake of the content.

Obstacles and Challenges

Throughout this process, I was faced with several obstacles and challenges.

Decentralizing both my role as ultimate authority and the role of the content held several expected struggles and some unexpected complications.

First, it takes time to foster a trust that allows students to be honest with a teacher.

Initial responses to survey questions and metacognitive exercises were met with cursory responses and shallow thought processes. Some students took the activities quite seriously and some did not. Inevitably, there were those students who were frustrated at their peers’ lack of engagement and other students who thought that the self-examination was a waste of time or foolish. The promise of anonymity can help assuage hesitation for some students, but completely anonymous feedback was not always the most helpful to me to foster a responsive environment. After time, though, all students became habituated to the metacognitive check-ins and, several times when they were not an explicit part of the lesson, would give me their thoughts, reactions, learning preferences etc. completely unprompted.

Second, students are not always accurate in their self-assessment. Initially projects were perceived of as “easier” than tests because the students didn’t have to spend the time studying. This translated, though, into a stylistic preference for the less complicated option rather than an educational preference for the best option. I was

frequently confronted with a student who had convinced themselves of a learning style preference or instructional preference, not because it was the most successful strategy, but because investment in the content was low and the technique or strategy was often one of a path of least resistance. Again, after time I noticed this shifted to more accurately reflect the strengths and weaknesses of the individual, however, the shift required much more one-on-one intervention. Resistance, at first, to new or complex strategies or techniques was more common than I expected and initially, students were happy to backslide into familiar routines, even if there was a measurable success with the new strategies or a noticeable lack of success with the patterned behavior. This process was slow in changing and can still be problematic if students are feeling a lack of confidence in the content of the class or are overwhelmed in other parts of their academic or personal lives.

Finally, relying on students to dictate the content of the course, duration of lessons and style of content acquisition and assessment lead to realm of complications to navigate. I had to be able to react and shift lessons quickly, meaning that the time spent researching areas of content and planning differentiated lessons was considerable. I had to have a loose design of a handful of lessons in every possible topic the students could select. This meant that I had to prepare lessons in subjects that may never be selected.

Each lesson, even those unused, had to be crafted to centralize the student experience and draw upon DI, attend to learning styles and draw creative inspiration from MI ideals.

The pressure to produce perfect lessons, differentiated with particular styles and intelligences in mind, required an immense amount of initial preparation. The time requirement lessened, though, as the students became more accustomed to the flow of the

class and were able to identify and then differentiate their own lessons. Additionally,

Survey of Advanced Topics has the added challenge of not being completely duplicatable. Since each group of students will select a different series of topics of study, each year I teach the class, I can recycle only some of the lessons.

I also had a particular challenge finding a balance between wanting the students to succeed and allowing them to consciously fail and learn from the process. If the challenge to the students is to create a grading rubric and then apply it to their own work, how could I respond to a student who failed the task? Periodically I had to resist the temptation to guide the student onto the right path if it was, in fact, my path instead of the student path. Part of the metacognitive experience, though, that tied together with a growth mindset, was ensuring the student knew the expectations and then allowing the students to fail. If they failed, once they failed, it was imperative that the students faced their failures, examine the problems and then devise specific solutions to fix the mistakes.

I was horrified when nearly half of the students failed their first presentation that followed student-created guidelines. Without exception, though, the students were able to identify their short-comings, acknowledge their own role in their lack of success and create a pathway to success the next time.

Role of Technology

I did not specifically set out to make a technologically integrated classroom.

Although technological scaffolding is frequently cited as a method to differentiate instruction, I found most examples to be content specific. Online practice problems, video tutorials or leveled homework trackers all hold content to be central and differentiate primarily on readiness levels, speed of content acquisition and ability to

master a skill. I, therefore, decided not to differentiate with technology as a central driver. In fact, most of my lessons specifically did not rely on technology (apps, computer programs, calculators etc.) in order to acquire knowledge, differentiate for challenge or scaffold skills. Technological integration, however, proved to be invaluable to the classroom in very surprising ways. Technological integration and programs such as Google Forms and Padlet, were an immediate way of eliciting feedback and gathering information from my students. The feedback loop with the students, the material in the class and the teacher was invaluable to the success of this course (Urbina-Lilback 2016 pg. 131).

I would like to speak, specifically of the utility of Google Forms. Information can be gathered anonymously or with identifying information. Information is tabulated in real time and can be used to simultaneously gather and display data. I found myself relying on this for establishing whole class confidence levels and measuring assessment readiness. High levels of confidence may indicate readiness for a summative assessment while low levels of confidence may indicate a further need to differentiate information acquisition or create an alternative formative assessment. Splits in confidence level may indicate a lack of interest, relation or engagement in the material or may indicate that the level of material is only reaching those students who have an affinity for the topic.

Google Forms also allows users to gather information publicly, by asking for the name of the student, or privately, by collecting the email addresses of the students as they respond. Individuating information and then attributing answers to a student may allow the teacher to intervene with a student who is silently struggling. On the other end of the spectrum, it also allowed me to identify the student who were consistently bored or not

reaching the limits of their potential. It may also establish a low stakes way for the student to communicate information to the teacher. The appearance of anonymity allowed the students to respond honestly, while the email identifiers allowed me to subtly shift structures of the course to meet specific needs of the class. For example, fast student feedback let me know that many of my students appreciated the template notes but needed a digital version accessible. They were being forced to take notes in their non- preferred fashion because of an oversight due to my own note-taking bias; I do not prefer to take notes digitally, so I had not given them a digital option. I started posting templates on a digital platform and met with a handful of students to talk about note- taking strategies in general. I made sure to meet with every student who had mentioned a struggle but frame it in a way such that none felt overwhelmed or over-scrutinized. With student feedback, I was better able to differentiate the instruction so that it fit both the needs of the class as a whole and the students as individuals.

Google Forms also made metacognitive reflections, goal setting and personal interaction quick and easy. I was able to establish baselines of student perception of success, overall confidence, and level of interest or engagement in the material. I was often surprised at introspection the students engaged in and, after the tasks were habituated, the level of self-reflection and self-honesty in which the students were able to engage. I was able to keep an eye on low confidence students, especially those whose level of self-perceived success was much lower than their quantitatively measurable level of success.

After the students requested more group work, I was also able to use Google

Drive to help differentiate the group work experience. Group work can always be a

challenge for teachers and students alike. Students can choose their own groups, which tends to give preference to students with strong social ties or strong interpersonal skills.

Students may not select scholastically strong groups and students who struggle with social interaction or have weak English language skills are often passed over, perpetuating the social or linguistic isolation. Teachers can also assign groups randomly which can stress students who are placed in groups that are not strong personality fits or fail completely due to underlying social issues or stratification not immediately apparent in a classroom. Observant teachers are aware of this Scylla and Charybdis of group work and do their best to balance the problem.

I used Google Drive to help navigate this experience. I asked each student to list three people they would prefer to work with and three people they would prefer not to work with. Students without strong preference for their interpersonal interactions could indicate this. On a practical level, this allowed me to uncover otherwise hidden social structures and stresses in the classroom. Students who appeared to be friends and work well together may not put each other on the “yes” list Students who would not be given the opportunity to self-select into groups would put each other on the “yes” list.

Additionally, it allowed me to identify students who consistently appeared on the “no” list and work with them and their classmates to reintegrate them into a working collective classroom experience. Over time, I noticed that the students who had chronically been on the “no” lists appeared less often, and students were more likely to choose best fit partners or groups rather than socially stratified groups. I also noticed that by asking student preferences, honoring some requests and actively addressing interpersonal skills, students were more likely to work together without friction, less likely to complain and

more likely to feel like to work had been evenly and fairly distributed. The process of integrating student opinions again altered the students’ approach to the task. They felt like their opinions had been considered and, if not honored, at least addressed.

Technology allowed me to record and track socioemotional tension and considerations in the classroom, address them and help dispel them.

Padlet is another program that I found to be particularly useful in the classroom.

Padlet acts as a digital bulletin board. A student can respond to prompts using photos, videos, words, text, links to websites, lists, gifs and more. The program encouraged interpersonal conversation between students and made metacognitive exploration simple.

I used Padlet when discussing grading, rubrics and expectations for assessments.

Students would brainstorm ideas for a grading system, recorded in Appendix 6, and then use the collected brainstorm to organize a coherent system. Student responses were refined and then directly used to create grading criteria. Using rubrics and a grading template, I was more easily able to create a student-centered system. Because the students had helped create the system, they were more able to understand the expectations and understand why assignments and assessments were graded in a certain way. Students who were struggling to meet expectations were able to find a path that allowed them to clearly demonstrate their knowledge and use their strengths to succeed.

Students who were meeting expectations were given a pathway to succeed or given the leniency to work at their preferred level when social, emotional or other academic pressures caused tension. Student who were excelling could mark their progress and student who were interested in challenging themselves had a clear pathway to do so. By allowing, encouraging and then integrating student feedback into the grading process,

each student had a clear idea of expectations, a clear reason for the grades given on an assignment or assessment and ownership over the material. As the year progressed all project, presentation and assessment were of a higher caliber, more accurately reflected the understanding of the student and more accurately reflected the individual interests, personality, creative strengths and intellectual skills of the student.

I also used Padlet as a way to link individual creativity to mathematical concepts.

Students who had preferences in visual communication expression, did not feel confident in their English language skills or did not have strong interpersonal communication skills were able to participate in the conversation without the strict need for verbal or written communication. This allowed students with different expression preferences to participate in conversations while staying in their comfort zones. The multimedia bulletin board, additionally, breaks open the ideas that mathematics is a sterile, calculation only, field and works to integrate the creative potential of an MI framework into a collective space.

Technological integration proved to be invaluable in my classroom. Even when not specifically involved in differentiation, technological platforms allowed students to build trust, interpersonal skills, ownership over the class and materials and a greater confidence in the processes of differentiation.

Applications in Other Classrooms

I started this project with an explicit focus on differentiation and gave it a central role in the classroom. Each aspect of this class design, from the subjects in the class itself to the lesson plans and student interaction was specifically chosen to create an interdisciplinary approach to mathematics that encouraged students to link their particular

creativity, intelligences and identity/culture to a mathematical platform. I acknowledge that this freedom is not possible in every classroom. I am particularly privileged to work at a school that encouraged the design of this class and, as an independent school, is largely free from state testing and other state or federal oversights. As a holistic concept, aspects of this class may not be available to all teachers.

There are several aspects, however, that I encourage every teacher address. First, as a metacognitive exercise, I recommend each teacher reexamine their curriculum with an eye towards differentiation as a core skill. Decentralizing the content, even for short periods of time, lay bare strengths and weaknesses and refocus the educational experience on the student. Rather than asking what am I teaching, ask how am I teaching it and how are the students learning. Rather than ask what content specific checkpoints the students are meeting, ask if they understand the best methods to meet a goal or objective. Rather than measuring the quantity of content acquired, ask in the student can present that content in a quality way. Rather than adding creativity to a lesson, ask how the creativity can inspire the acquisition of the material itself. Teachers should examine their own biases towards the learning experience, which can also be done by centralizing differentiation. Ask, as a teacher, am I giving notes in a certain way because I prefer notes in a certain way? Do I have cultural or linguistic biases that I project on my students? Do I expect my students to learn the way that I learn? Am I actively exploring and exposing my students to a variety of methods of content acquisition?

For those teachers who have the freedom to design creative or differentiated assessments, I encourage this as well. No matter the type or level of classroom, assessment must occur. Ask, I am creating assessments that accurately measure a

student’s content acquisition? Am I considering the preferences of the student? Am I taking into account the ways that confidence and anxiety can impact an assessment? Am

I teaching in a variety of ways but assessing in only one way? Do my methods of assessment prioritize or favor one type of student? Do I have cultural or linguistic biases that I am unconsciously projecting onto my students? Am I missing opportunities to elicit responses from marginalized groups of students? Am I giving all aspects of intelligence an opportunity to shine? It is not always possible in all classrooms in all school systems to radically alter lessons and assessments. It is possible, however, to offer students multiple access points to demonstrating knowledge.

I recommend that teachers, in reexamining their curriculum, interrogate the ways they are considering the socioemotional health and identities of their students.

Mathematics is often taught as an abstract skill, removed from social, historical or cultural context. By reexamining curriculum with a student-centered lens, a teacher can analyze if this categorically must be the case. Start by asking if the identities of the students are reflected in the classroom and lessons. Ask, is the diversity of my student body represented in my curriculum in a way that is foundationally substantive? Do I have a variety of role models from the past and present for my students to follow and do I address these people purposefully and by name? Do I explicitly address issues of equity and diversity in the classroom? Do I acknowledge the contributions of mathematicians from a variety of cultures, ethnicities, sexualities etc.? Do I humanize or contextualize mathematical concepts? Do I make sure my students can see their own identities reflected in the content?

Apart from connecting the students’ identities within a mathematical and historical context, teachers can also reexamine role that student mood and affect can play in lessons and performance. Ask, do I verbally acknowledge the ways in which confidence and anxiety can affect performance? Do I specifically address the needs of non-native English speakers? Do I actively foster safe environments for academic discussion and exploration? Do I design lessons that encourage collaboration and interpersonal skills? Do my lessons integrate the needs of students with a variety of processing speeds and abilities? In reexamining the curriculum through the lens of the student experience, identity and socioemotional well-being, teachers may be able to identify where their goals are not meeting the reality of the educational experience. If students are expected to collaborate and communicate but the lesson does not identify or teach, students who excel in these areas will continue to excel and students who struggle in these areas will continue to struggle. Seldom does a teacher enter the profession without wishing the best for the holistic education of the student. But seldom in a math class are linguistic or collaborative skills purposefully addressed and directly framed within a lesson and taught as distinct skills.

In many schools, the requirements of the class, school district or state are such that a structural rearrangement of the course is not possible. In these cases, the attention to metacognitive structures and processes can still be integrated into the classroom. Ask students how they prefer to take notes and then, if possible, respond by altering the method of information acquisition. If not possible, respond by acknowledging and justifying a certain way of content acquisition. Ask students if homework is the best way for them to practice. Acknowledge the responses and whenever possible, alter parts of

the class to address student input and perceptions. Ask students about their preferred method of assessment. Ask students about their strengths and weaknesses. Asking consistently, acknowledging their responses and fostering an expectation of honesty and interaction with the process of learning fosters an atmosphere of trust within the classroom. I was interested in the outcomes of this part of my class because reliably asking the students for their preferences and input and then explaining why I could or could not adhere to their requests made every task more likely to succeed. When the students asked to be given oral exams, rather than written exams, I could not honor this request. A brief discussion at the beginning of the class about the reasons I would be giving a written exam was enough for the students to recognize the importance of the assessment and perform to the best of their abilities. The act of asking and then acknowledging their requests, even without complying, was paramount. Involving the students in the process of creating rubrics and grading systems ensures the students understand expectations, feel invested in the outcome because they were a part of the process and gives them multiple pathways to success or excellence, even if they do not actually succeed. Opening the process of assessment creation and grading also allows the teacher to decentralize themselves and sole curator of knowledge and helps the students cultivate a sense of curiosity about and authority over a topic. Giving the students a chance to lead seemed, paradoxically, to make them more willing to enthusiastically follow when necessary. My students were more willing to engage in traditional top down instructional methods when they knew it was not the only method available in the class.

The feedback loop between students and teachers can be used in any classroom and fosters trust. Trust fosters engagement and students who engage with the material,

even when they find it complex or boring, are more likely to succeed at the task.

Feedback, digital or otherwise, can be used in any classroom to ascertain assessment readiness, skills confidence, gaps in knowledge and underlying social struggles that may be interfering with the classroom environment. Quick check-ins about a variety of topics can be quickly integrated into any classroom and, in fact, was so successful in this class, that I have started to use it in my more traditional classes. In those classes, too, students seem less resistant to non-preferred tasks because they feel less like receptacles into which knowledge is being poured, and more like individuals who can interact with their own knowledge demonstration and content acquisition.

Future Considerations for the Class

This experiment in educational process has been quite inspiring. I would like to take what I have learned and continue the work in two concrete way. First, I would like to apply some of the collaborative and metacognitive work to my more traditional classes. Although I will not have the holistic control over the specific content and will not be able to ask the students to dictate the course of the content (our school follows a curriculum very similar to Core Curriculum) I can integrate the student involvement in areas of rubric creation, confidence checks, and assessment by allowing the students to choose the way they best demonstrate their knowledge. I would like firsthand experience determining if the best practices work in my own more traditional classrooms, how much and in what ways.

Second, I would like to take interdisciplinary work that I have started in this class and continue it within the school. Since much of my curriculum depends on providing a historical context, I would like to work with the History Department to form meaningful

lessons that can explore best practices and concepts from both classes. I would like to work with the Art Department in a foundational way to help with the creation of project- based rubrics, non-trivial art projects and gallery quality mathematical and artistic work.

If one of the stated purposes of this class is to historically contextualize the discipline of mathematics, I would like to consciously and purposefully reach out and create a true interdisciplinary program.

Conclusion

This class was an interesting and illuminating exploration of Differentiated

Instruction and the ways it can transform a mathematics classroom. I started the project to discover if, with a creative curriculum inspired by a Multiple Intelligences platform, I could provide a contextually more applicable and rewarding differentiated mathematics experience for the students. Over the course of two years of teaching this course and several months of asking the students for feedback about each aspect of differentiation, I discovered that, while the focus on differentiation had positive effect on me as a teacher and the way that I conceptualize instruction and assessment in the classroom, the more important overall alteration to the curriculum were the metacognitive elements.

Involving the students in the process of choosing the direction of the course, the methods of instruction and the methods of assessment made the students more invested in the class and more likely to perform well at all tasks, even those tasks outside of the students’ self- reported comfort zone. Asking the students to analyze their own thought processes and preferences for instructional techniques and assessment styles allowed each student to identify their strengths and weaknesses, perform well within those categories and work to build up weaknesses in non-preferred categories. Finally, asking students about the role

of confidence and anxiety on assessments normalized the feelings of anxiety that students often face. Using technological platforms to ascertain underlying structures that could lead to social or emotional stressors in group and collaborative work lead to more effective working groups and the ability for me to care for not only the academic but also social needs of my students. Consistently focusing on the way my students race, ethnicity, language, gender and/or sexuality were or were not reflected in the mathematics made each student be able to see their own identities reflected in the abstract subjects. I hope to continue to build upon the ideas of this class and study to make a more interdisciplinary classroom experience and continue to encourage a fully integrated, student-centered, metacognitive experience.

Appendix 1

Lesson 1: Information Acquisition

I have reproduced the worksheet in a slightly condensed form. Usually students would be given ample space to show their work on the page itself.

Changing Bases: Template Notes and Individual Work

Changing Bases: Notes and Individual Work

Hindu-Arabic number system that we use is a base 10 system. We can imagine each position like a cup. When we fill the cup, we move over to the next position and start the process again. Remember, a base 10 system uses 10 digits (0 through 9). Then number 10 has two different digits, a 1 and a 0.

Example:

5467

Thousands Hundreds Tens Ones

5 4 6 7

Because we count in a base 10 system, we can write a breakdown (or decomposition) as powers of 10

Thousands Hundreds Tens Ones

5000 + 400 + 60 + 7

5 x 103 4 x 102 6 x 101 7 x 100

Changing from other bases to base 10

Other bases operate the same way. Instead of the “filled” position occurring after 9, it occurs after the base minus one. This means that a base 5 system “refills” after 4, a base 8 system “refills” after 7, a base two system (also called binary) “refills” after 1. Bases higher than 10 get tricky because we need a unique symbol for numbers above 9. The number 10 needs a unique symbol, as does the number 11 and so on. When we count in base 12, for example, typically 10=A and 11=B.

Like above, we can use the power notation to convert from different bases into base 10.

Examples: In base 5,

3 4 2 2

3 x 53 4 x 52 2 x 51 2 x 50

375 + 100 + 10 + 2

So, 3422 in base 5 is 487 when we convert to base 10.

NOTICE: We are using 5some number instead of 10some number . This is because we are working in base 5 instead of base 10.

SOME MORE EXAMPLES:

In base 2 (binary), 1 0 0 0 1 0 1 1

1 x 27 0 x 26 0 x 25 0 x 24 1 x 23 0 x 22 1 x 21 1 x 20

128 + 0 + 0 + 0 + 8 + 0 + 2 + 1

So, 10001011 in binary is 139 in base 10.

NOTICE: We are using 2some number because we are working in base 2.

Now let’s try challenge mode. Base 12! Remember, when we work with bases bigger than 10, we need a unique symbol for 10, 11 etc. Generally, we use A for 10, B for 11, C for 12 etc.

In base 12 3 B (remember, this is 11) A (remember, this is 10) 1 9

3 x 124 11 x 123 10 x 122 1 x 121 9 x 120

62,208 + 19, 008 + 1,440 + 12 + 9 +

So, 3BA19 converted to base 10 is 82,677

Try the following on your own. Convert all numbers into base 10.

1. Starting in Base 6, convert to base 10 12455

2. Starting in Base 3, convert to base 10 12200

3. Starting in Base 12, convert to base 10 21BB

4. Starting in Base 8, convert to base 10 41177

5. Starting in Base 2, convert to base 10 10110001

Challenge: What happens if you have a fraction or decimal in a different base? How would you change to base 10? Try with the following:

6. Starting with Base 4, convert to base 10 𝟏

𝟐

7. Starting with Base 15, convert to base 10 (Remember, in this case, A=10, B=11, C=12, D=13, E=14)

6 퐴

Changing from base 10 to other bases

In order to change from base 10 to other bases, we need to remember the first part of the notes. We originally ADDED and MULTIPLIED to convert from other bases to base 10. If we want to reverse the process, we use the inverse operations. Since we originally added and multiplied we DIVIDE and SUBTRACT to reverse the process.

FOR EXAMPLE:

1. Convert from 392 (in base 10) into base 3.

a. First, we need to figure out the values of the base positions i. 1 x 30 = 1 ii. 1 x 31 = 3 iii. 1 x 32 = 9 iv. 1 x 33 = 27 v. 1 x 34 = 81 vi. 1 x 35 = 243 vii. 1 x 36 = 729

Our number, 392, is between 243 and 729. If we remember the bucket analogy, we are looking at the nearest partially filled “bucket”. In our case, this means that we will have 6 places in our base system.

b. Then, we figure out the values using subtraction i. Start by subtracting the nearest base value from your starting value. 392 – 243 = 149. Since the answer is less than 243, we know we have a 1 in the 243rds place (a 1 in the 5th position) ii. Continue in this fashion 149– 81 = 68 (1 in the 4th position) iii. 68 – 27 = 41 41 – 27 = 14 (I had to subtract twice, so I have a 2 in the 3rd position) iv. 14 – 9 = 5 (1 in the 2nd position) v. 5 – 3 = 2 (1 in the 1st position) vi. 2 – 1 = 1 1 – 1 = 0 (2 in the 0th position)

c. Finally, we can conclude that 392 (starting in base 10) is 112112 when we convert to base 3

ANOTHER EXAMPLE:

2. Convert from 392 (in base 10) into base 5. a. First, we need to figure out the values of the base positions. i. 1 x 50 = 1 ii. 1 x 51 = 5 iii. 1 x 52 = 25 iv. 1 x 53 = 125 v. 1 x 54 = 625

Our number is between 125 and 625 which means we will have 4 places in our base- changed number

b. Then, we figure out the values using subtraction (Challenge: how would you use division instead of subtraction?)

i. 392 - 125 = 267 267 - 125 = 143 143 - 125 = 17 (3 in the 3th position) ii. 17 - 25 = Yikes! A negative number! (Don’t worry! This means that there is a zero in the 2nd position because nothing fulfills the criteria) iii. 17 - 5 = 12 12 - 5 = 7 7 - 5 = 2 ( 3 in the 1st position) iv. 2 - 1 = 1 1 - 1 = 0 (2 in the 0th position)

c. Finally, we can conclude that 392 (starting in base 10) is 3032 when we convert to base 5.

Try the following on your own. Convert all numbers from base 10 to the indicated base.

1. Convert 548 (base 10) into base 9

2. Convert 849 (base 10) into base 3

3. Convert 6721 (base 10) into base 7

4. Convert 99 (base 10) into base 2

5. Convert 761 (base 10) into base 12

Challenge: What happens if you want a fraction or a decimal in a different base? How would you change from base 10? Try with the following:

6. Starting with base 10, convert the following into base 5 1 2

7. Starting with base 10, convert the following into base 15 7 10

Appendix 2

Lesson 2: Summative Assessment Text Resources

I have reproduced the assessment text resources in a slightly condensed form.

Usually students would be given ample space to show their work on the page itself.

Preparation for Summative Assessment- Study Guide

1. Name some example of “animal math. Do you think this demonstrates

mathematical understanding? Why or why not?

2. What was the first record of human counting?

a. How old is it?

b. Where was it found?

c. What number systems does it use?

d. How do we know this is counting?

3. What is the first record of human calculation?

a. How old is it?

b. Where was it found?

c. What number system does it use?

d. How do we know this is calculation?

4. What is the difference between and additive and a positional system?

a. Give one historical example of an additive system. How do we know it is

additive?

b. Give one historical example of a positional system. How do we know it is

positional?

5. Show and solve 5 + 8 in the Tally System

6. Show and solve 345 – 212 in the Egyptian System

7. Show and solve 780 + 23 in the Mayan System

8. Show and solve 5436 – 1900 in Roman Numerals

9. Discuss what makes one of the following number systems unique: Modern

Chinese Characters, French, Cuneiform

10. Name two examples of Physical Counting Systems. How are they similar. How

are they different?

Summative Assessment- Traditional Style Test

Confidence Check and Metacognitive Activity

Before you start this quiz, take some time to engage in some metacognition. Write a few sentences about your confidence level. How do you feel? Prepared? Did you study? How did you study? Are you a nervous? Unsure? Excited? Think about your own thinking!

Short Answer: History

Choose two of the following questions to answer. Answer all parts of the question. Use complete sentences. (10 points)

1. Ants have been found to count their steps using an internal odometer. Does this

constitute mathematical understanding? Why or why not?

2. The Lebombo bone and the Ishango bone are both examples of very old records

of human counting. One, however, is widely regarded as being the first and the

other is still under debate. Which one is seen as being “the first” example? Why

is this?

3. (Challenge) The Rhind Papyrus and Plympton 322 are both landmark documents

in the history of mathematics. What are they and why are they important?

Number Systems: Calculation

Choose 3 of the problems below to answer. You may use modern signs if necessary.

Demonstrate your knowledge by thinking like an ancient person. Try not to use your modern thinking! (5 points each)

1. Solve 8756 - 319 using the Egyptian Number System

2. Solve 561 + 87 using Roman Numerals

3. Solve 12 ÷ 3 using the Tally System

4. (Challenge) Solve 70 + 702 using the Mayan Numerals

Long Answer: Number System

Answer all of the following questions. Answer all parts of each question. Use complete sentences.

1. What is the difference between an additive and a positional number system? Use

historical or modern examples to back up your reasoning. What are the benefits

and drawbacks to each type of system? (10 points)

2. Physical Counting Systems is a broad category that can include Body Numbering

Systems or Gestural Systems. Discuss an example of each of these types of

systems. What are the benefits and drawbacks to each type of system? (10 points)

Critical Thinking: Mathematics

What is mathematics? Is it a human invention? Does mathematics exist in the outside world and humans discover it? Why do we study mathematics as a separate discipline? What makes it different than other disciplines? Use class discussions and any readings to create a working definition of mathematics. (15 points).

Summative Assessment- Project Text

Congratulations! You can count! And now, you can count in Egyptian, Mayan,

Babylonian, Chinese and ASL. Some number systems don’t exist anymore (Cuneiform), some have changed quite a bit over time (Hindu-Arabic). Some are very complex

(Aztec) and some are very simple (Tally). With all of this knowledge, it is time to re- invent the wheel…or zeros.

The Assignment:

Invent your own number system. It can be physical, oral, or written. If it is written, it must be positional or additive.

Write a paragraph explaining your number system.

Create an artifact demonstrating your number system in action. Depending on the number system you invent, your artifact will have different forms.

Appendix 3

Lesson 3: Formative Assessment Student Directions and Assignment

I have reproduced below the student-facing assignment and details. It has been slightly condensed to remove personal identifying features (student names, dates etc.) and to remove the space given to the students to brainstorm or write ideas.

Project Text- Introduction to Assignment

Gallery of Codes Mini-Project

You are curators of the Gallery of Codes, a very elite educational institute. As a group you must delight and inform your spectators about your code.

1. As a group research your code. Be sure you understand:

a. HOW the code works

b. WHO wrote it

c. WHY or the context in which it is or was used

d. WHEN it was made

e. WHAT kind of code it is (steganography, substitution,

transposition etc.)

f. Any other important or interesting information

2. Create (as a group or assign roles) an informative poster explaining your code.

Make sure the information is

a. Neat

b. Clearly explained

c. Creatively arranged with Visual Aids and Colors

3. Choose a member of the group to be the Expert Explainers and the others to be

Traveling Scholars.

a. The Expert Explainer will stay with the poster and be prepared to

talk with other students and explain the details of their code.

b. The Traveling Scholars will journey around the exhibit, learning

about the other codes. They must be prepared to return to their

Expert Explainer and teach about the other codes in the Gallery.

Appendix 4

Lesson 4: Information Acquisition Notes Template

I give each student the option to use this template, which I post online for the students to access as they need. For those students who struggle with executive functioning or organizational skills, I print the template and give it to them specifically.

The Topics addressed in the template change as needed.

Table 5. Notes Template Topic #1 Topic #2 Topic #3

Turing- The Man Turing – The Machine Turing – The Mathematics

Fact

Source

Fact

Source

Fact

Source

Fact

Source

All Templates for notes that I assign as homework are a variation of this format. It is easily accessible in a digital format, allowing students to handwrite the information or type the information as needed.

Appendix 5

Text of Student Surveys

I have recreated the text of all student survey, as well as some contextual information.

Differentiated Instruction as it pertains to Content Acquisition and Instruction Technique

This questionnaire was given via Google Forms. It was administered during the introduction of the topic but before any summative assessment. This survey was given once during the unit. I revisited this questionnaire again at the beginning of the new term. The survey asks about the students’ preferred method of content acquisition.

Specific class activities were not altered according to student response during the first term. During the second term, introductory content was given in the fashion most requested. Responses were aggregate.

1. What type of classroom instruction do you [the student] prefer?

a. Lecture only.

b. Lecture with notes.

c. Hands-on activities.

d. I don’t like classroom instruction. I would prefer to teach myself.

e. Other. Please describe:

2. What type of notes do you prefer to take?

a. Notes with a template.

b. Notes without a template.

c. Notes from the white board/projector

d. I don’t take notes. I remember everything.

e. I don’t know what type of notes I prefer.

Differentiated Instruction as it pertains to Assessment

This questionnaire was given via Google Forms. It was administered after the introduction of the topic but before any summative assessment. This survey was given once during the unit. Specific assessments were not altered according to student response during the first term. During the second term, students were allowed to select their preferred method of knowledge demonstration for at least one summative assessment.

Responses were aggregate.

1. How do you prefer to demonstrate your knowledge?

a. Test or Quiz.

b. Presentation in front of the class.

c. Poster or other visual display.

d. Project-based or applied work.

e. Oral exam or conversation with peers or teachers.

f. Artistic option (Dance, Skit, Performance Art etc.)

g. Academic paper or written short answer.

h. Other. Please describe:

2. Why do you prefer this type of assessment?

3. Do you always prefer this type of assessment? Why?

4. What is your least preferred style of assessment? Why?

Differentiated Instruction as it pertains to Confidence Level and Mood

These questionnaires were given via Google Forms. The first was administered after the introduction of the topic but before any summative assessment. The second was administered immediately before the summative assessment for the unit. They may be administered during any time, however. These questions may be useful to assess readiness before the formal introduction of a unit. They may also be useful during the unit but before any summative assessment or immediately prior to a summative assessment to gauge student confidence or readiness. These surveys ask students about their confidence level and explores its impact on successful demonstration of knowledge.

Questions on the first survey were subject specific and may be altered as the unit changes. Responses were aggregate during the first term. During the second term, I collected specific information on specific students.

(Survey 1. Assessment of comfort level as it pertains to knowledge acquisition)

1. I feel comfortable with (insert applicable skills)

2. I need more practice with (insert applicable skills)

3. I don’t understand (insert applicable skills)

(Survey 2. Assessment of mood and confidence as it pertains to performance)

1. Do you worry or feel anxious about assessments?

2. Do you think your success is influenced by your confidence?

3. What makes you feel most worried about assessments?

4. What makes you feel less worried about assessments?

5. (If you have test anxiety) Do you think the assessment [test, oral exam, project,

presentation etc.] is a fair way to measure your knowledge? What makes it fair or

unfair? What would make this assessment as fair as possible?

Differentiated Instruction as it pertains to Content of Course

These questionnaires were given via Google Forms. The first was administered after the summative assessment of the previous unit but before the introduction of the new unit. The results of the survey directly dictate the specific content of the unit.

Questionnaires were altered over the course of the terms to reflect new topics and eliminate previous topics of study. Responses were aggregate.

The second was given after the introduction of the new topic but before any assessment, formative or summative. Specific class activities were not altered according to student response during the first term. During the second term, rubrics and grading criteria were altered according to class consensus. Responses were aggregate.

The third questionnaire was given after any project, group or individual, or formative or summative assessment. Responses were not aggregate. Specific information was collected on specific students.

The fourth questionnaire was given after any project. Responses were not aggregate. Specific information was collected on specific students. This question was meant to help foster a sense of reflective empowerment in the work and help establish a consensual library of exemplars for future students.

(Survey 1. Dictation of direction of class and content of study)

1. Where do we go from here? (Give list of topics and then “Other” option).

2. What interests you about the topic?

3. Are you interested in this topic outside of class?

(Survey 2. Creation of Grading Rubrics)

1. Class Participation

a. What is class participation?

b. What indicates “good” class participation? What indicates “bad” class

participation?

c. What should the teacher grade as “class participation”?

d. How many points (if any) should be associated with class participation?

2. Class Presentation

a. What makes a “good” presentation? What are some key features of a

“good” presentation?

b. What should the teacher grade in presentation?

c. What points should be associated with key features in a presentation?

3. Projects

a. What makes a “good” project? What are some key features of a “good”

project?

b. What should the teacher grade in a project?

c. What points should be associated with key features in a project?

(Survey 3. Self-Reflection)

1. Name

2. Group Member Names [excluded if the project was individual].

3. Topic [or Project Title]

4. What grade do you think you earned for the project? Why?

5. What grade do you think your group earned for the project? Why? [excluded if

this project was individual].

6. What went well? What are you proud of? What worked well? What was

interesting?

7. What did not go well? What was problematic? What was the biggest challenge?

What do you wish you had done differently?

8. Any other comments or observations? [Optional]

(Survey 4. Exemplars of work)

1. Are you proud of your work? Why or why not?

2. Do you feel comfortable with your work being used as an example to future

students?

Appendix 6

Student Responses to Selected Survey Questions

Below I have recreated the responses to selected survey questions. Responses have been edited slightly to remove personal identifying information (names, ages etc.) and when spelling errors inhibit clarity of response.

Student Response to Preferred Assessment Style

Students were asked the question, “How do you prefer to demonstrate your knowledge?” and then several clarifying questions.

Figure 6. Student Response to Assessment Style

Student Response to the question: How do you prefer to demonstrate your knowledge?

Why do you prefer this type of assessment?

• It tends to show more of what I have learned

• It is easier to show everything I have learned and there is a set rubric to follow

• I prefer project-based because I am a hands-on person and can be successful when

I don’t save my work until the night before. Projects usually do not create the

stress that tests and quizzes do.

• To me this best shows what I know and understand and what I found interesting

in this topic

• More interesting

• It’s something that eases the tension on work because working with a partner

divides the work and the stress into segments

Do you always prefer this style of assessment? Why or why not?

• Not always. Sometimes I prefer a poster or other visual display. Either way, it

demonstrates to my teachers and peers what I have learned.

• Not always. Sometimes I do better on tests of I would rather get the assessment

done in one sitting. Posters I have to work on for a couple of days and if I happen

to turn it in late, I get a lot of points off.

• I usually prefer this style of assessment because I do not always do well on tests.

Testing and studying have never been my strengths. A project gives me a better

opportunity to influence my grade and show the information.

• Almost always. I think for me it is the best way my brain can process and show

what I know and understand.

100

• Yeah, because it interests me.

• I don’t always prefer this style of assessment. Sometimes a quiz can be useful if

it is for the benefit of the student and not so stressful. Other times a visual

presentation is a good way of expression without words, which many students can

find difficult.

What is your least preferred style of assessment? Why?

• Tests/quizzes because I am a pretty bad test taker.

• I don’t like skits or talking in front of the whole class.

• Tests and standing up in front of the class are my least preferred style of

assessment because I don’t have a lot of confidence in myself and I personally

don’t like tests.

• A test. I have very bad test anxiety and get really nervous and psych myself out a

lot.

• Quiz. It makes no sense to take a quiz in this class.

• Oral exam. Most students, including I, have difficulty translating thoughts into

speech that makes sense. To me, I know what I am talking about more if I am

writing than speaking.

Student Response to Question: What makes a “good” project?

Students responded to this question via Padlet, then worked together to organize the information into cohesive categories.

101

An “A” Project:

• An “A” project includes neatness, clearly shows that you know the concept that is

asked for you to present and you make it in a creative way.

• An “A” project, in my opinion, not only follows the instructions and does what

the rubric says, but also gives some individual identity to it. For example, if I

wanted to write a math equation, I should probably draw in bright identifiable

colors or fonts. Not too much so that it distracts from the assignment, but enough

to make it interesting and get people reading it. It is finding the perfect middle

between just getting it done and getting carried away making it perfect.

• An “A” project should follow all of the instructions on the rubric. It always has to

be “attractive” and explain everything well. Everything has to make sense

including title, pictures and words. Also, it have to be clean and clear.

• An “A” project is creative, clean, cute, neat, it follows the instructions perfectly

and shows that the person knows exactly what they are talking about. It shows

there was a lot of effort put into it.

• An “A” project should go the extra mile as in making a video or doing something

that can be called extra in order to get that “A”.

• An “A” project should go above standards that the teacher sets and be extremely

organized, creative and capturing to the mind of the person grading it.

• An “A” project should go above and beyond what the teacher expects and be

creative and exciting.

• An “A” project should tell readers a good explanation of the topic. Also it get

interest from people and be creative.

102

A “B” Project:

• A “B” project should explain what the topic is about and that’s about all.

• A “B” project should meet all the standards but not be as creative or mind

blowing.

• A “B” project shows that there was effort put in but didn’t go above what was

expected. It follows the instructions but not entirely and it is neat but not as neat

as you know it could be.

• A “B” project is you do the steps that the teacher wants but you don’t put as much

effort into the project so it is not as neat as it should be.

• A “B” project should be a little worse than an “A” project. Might not really be

clean or one or two parts that doesn’t make sense. But it should have mostly

everything.

• A “B” project should meet the standards that the teacher sets, but doesn’t do

anything new and exciting.

A “C” Project:

• A “C” project will meet some of the expectations but not all of them.

• A “C” project should meet just the bare minimum requirements of the projects

with nothing else.

• A “C” project must be missing something, like a paragraph, or just every part is

bad.

103

A “D or F” Project:

• A “D” project is doing almost all the work required but not quite

• A “D” project and “F” project are just bad and messy and missing a lot of parts of

the rubric.

• “D” and “F” projects won’t meet expectations and won’t be creative.

• An “F” project would be meeting only one of the expectations or requirements.

Other Responses

• A good project is one that clearly shows someone’s creative and deep thinking in

an attractive way. It is taking hard thinking and transforming into a nice visual

thing.

• A good project should include the elements of neatness, completeness and

creative. In order to meet this expectations, the project should have clear,

colorfully appropriate visual, as well as writing to explain and complete them.

The project should also express the teacher requirements in the rubrics or

directions stated

• A good project includes something that can be visual or organized to show to the

class or teacher. It must engage the audience for the entire presentation, not be

boring and have real and tangible evidence to support their claims.

• A good project should catch people’s attention with a hook. It should look neat

and easy to read. All the information should be organized and it should represent

to information and topic well.

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• A good project should be informative on the topic. You are presenting what you

know about a specific topic.

• A good project should have something that is interesting and attract people’s eye.

You want to make people know about the topic.

Responses to the questions about confidence and concern with assessments

1. Do you feel worry or anxious about assessments?

a. Yes. I doubt myself and I need to create better study habits but

don’t know how

b. Yes. It matters a lot and it’s a big part of my grade

c. Sometimes but not always

2. Do you think your success is influenced by your confidence? How?

a. Definitely not all of it. I sometimes doubt the correctness of my

answers or overthink things. I think good study habits are a key to

success.

b. Partly. I think we need to think we can succeed before we can

succeed.

c. Yes, very much so.

3. What makes you feel most worried about assessments?

a. Do I know everything I need to know? I’ll probably just fail so

why try? I didn’t study enough.

b. I might not have reviewed things or there might be things I still

don’t understand.

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c. When I can’t finish on time or I know I don’t know the answers to

some of the questions.

4. What makes you feel less worried about assessments?

a. I usually feel less worried about assessments when I know the

material well. I just really suck at studying

b. I focus on the things I do know

c. When I have studied I feel confident that I can answer most of the

questions on the test.

5. (If you have test anxiety) Do you think the assessment is a fair way to measure

your knowledge? Why or why not? What would make as assessment as fair as

possible?

a. Test Anxiety is very normal and healthy as long as it is minimal. I

unfortunately have a lot of anxiety. I feel the test is fair, though.

I’m not saying there is a more fair way of testing, but the test could

maybe offer more interactive ways to test our knowledge. Maybe

have some group sections where we work together to solve a

problem in a fun or interactive way. Provide a challenge of some

sort.

b. Not really because tests just make people feel bad and we forget it

after anyway. Tests are just kind of useless. Maybe projects are

better because we care and remember them.

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c. I really prefer projects because I am not a very good test taker.

However I really liked it when we got to choose how much the test

or project mattered. That helped my anxiety a lot.

Student Response to Preferred Classroom Instruction

Figure 7. Student Response to Preferred Instruction Style

Student Response to the question: What is your preferred style of classroom instruction?

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